Financial Derivatives 3
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these related forecasts. First I forecast the euro would strengthen as European economic recovery picked up and the US economy slowed. Second, I forecast euro strength would be augmented over the next five years by a reduction of Europe's $100 billion to $150 billion of excess dollar reserves. Third, I forecast that the authorities would be less concerned over exchange rates, would only intervene after bigger exchange rate moves, and so exchange rate volatility would rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director, Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Important Concepts in Chapter 1
n Different types of derivatives n Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency n Theoretical fair value n Arbitrage, storage, and delivery n The role of derivative markets n Criticisms of derivatives n Ethics
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Business risk vs. financial risk n Derivatives u A derivative is a financial instrument whose return is derived from the
return on another instrument.
n Size of the OTC derivatives market at year-end 2010 u $601 trillion notional principal u GDP is only $15 trillion u See Figure 1.1 and Figure 1.2
n Real vs. financial assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Derivative Markets u Over-the-counter and exchange traded u Exchange traded derivatives volume in 2010 was over 22 billion contracts
on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information
u Derivatives trade all over the world u See Table 1.1 for the top ten derivatives exchanges
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivative Markets and Instruments
n Options u Definition: a contract between two parties that gives one party, the buyer,
the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today
u Option terminology F price/premium F call/put F exchange-listed vs. over-the-counter options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Forward Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today u Exclusively over-the-counter
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Futures Contracts u Definition: a contract between two parties for one party to buy something
from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Options on Futures (also known as commodity options or futures options)
u Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today
u Exclusively traded on a futures exchange
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Swaps and Other Derivatives u Definition of a swap: a contract in which two parties agree to exchange a
series of cash flows u Exclusively over-the-counter u Other types of derivatives include swaptions and hybrids. Their creation
is a process called financial engineering. n The Underlying Asset u Called the underlying u A derivative derives its value from the underlying.
Derivative Markets and Instruments (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Some Important Concepts in Financial and Derivative Markets
n Presuppositions – rule of law, property rights, culture of trust
n Risk Preference u Risk aversion vs. risk neutrality u Risk premium
n Short Selling n Repurchase agreements (repos) n Return and Risk u Risk defined u The risk-return tradeoff (see Figure 1.3)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
n Market Efficiency and Theoretical Fair Value u Efficient market defined: A market in which the price of an asset equals its
true economic value. u An efficient market is a consequence of rational and knowledgeable
investor behavior u The concept of theoretical fair value F The true economic value
Some Important Concepts in Financial and Derivative Markets (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Fundamental Linkages Between Spot and Derivative Markets
n Arbitrage and the Law of One Price u Arbitrage defined: A type of profit-seeking transaction where the same
good trades at two prices. u Example: See Figure 1.4 F The concept of states of the world
u The Law of One Price
n The Storage Mechanism: Spreading Consumption across Time
n Delivery and Settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
The Role of Derivative Markets n Risk Management u Hedging vs. speculation u Setting risk to an acceptable level u Example: Southwest Airlines
n Price Discovery n Operational Advantages u Transaction costs u Liquidity u Ease of short selling
n Market Efficiency
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Criticisms of Derivative Markets
n Speculation n Comparison to gambling
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Misuses of Derivatives
n High leverage n Inappropriate use
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Derivatives and Ethics
n Codes of ethics and standards of professional conduct are vital components of the derivatives profession
n Examples u CFA Institute u Professional Risk Managers International Association u Global Association of Risk Professionals
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
Derivatives and Your Career n Financial management in a business n Small businesses ownership n Investment management n Public service
Summary
Source of Information on Derivatives http://www.cengage.com/finance/chance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 1: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
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Chance/Brooks An Introduction to Derivatives and Risk
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Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty. Eyes flitting between flickering lines of information on four different screens, one ear on the phone, the other on the cries of the colleagues, twelve hours of split-second calculations, judging yourself and being judged on the score at the end of every day. These men and women lived and breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
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Important Concepts in Chapter 2 n Definitions and examples of call and put options n Institutional characteristics of options markets n Options available for trading n Placing an options order n The clearinghouse n Accessing option price quotations n Transaction costs n Regulation of options markets n Margins and taxes in option transactions
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option terminology u price/premium u call vs. put u exercise price/strike price/striking price u expiration date
Everyday examples of options u rain check u discount coupon u airline ticket with cancellation right u right to drop a course
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Development of Options Markets
n Early origins n Put and Call Brokers and Dealers Association n Chicago Board Options Exchange, 1973 n Resurgence of over-the-counter market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Call Options
n Current example n Objective of a call buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Put Options
n Current example n Objective of a put buyer n Moneyness concepts u In-the-money u Out-of-the-money u At-the-money
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Trading Activity
n In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine)
n In 2010, over-the-counter option volume approximately $64 trillion notional principal and $2.2 trillion market value (Bank of International Settlements)
n OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1)
n OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Over-the-Counter Options Market
n Worldwide n Credit risk n Customized terms n Private transactions n Unregulated n Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Organized Options Trading
n Listing Requirements n Contract Size n Exercise Prices n Expiration Dates n Position and Exercise Limits
The concept of an options exchange
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Traders
n Liquidity Providers u Provide bid and ask prices to facilitate trading u Scalpers, position traders, spreaders u Lead market makers, designated primary market makers
n Floor Broker – acts as agent for customers n Order Book Official u Limit orders u Electronic order processing
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Options Traders (continued) n Other Option Trading Systems u Specialists u Registered options traders u Electronic trading systems
n Off-Floor Option Traders u Option brokers u Proprietary options traders
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading
n Placing an Opening Order u Types of orders
n Role of the Clearinghouse u Options Clearing Corporation (OCC) u Clearing firms u See Figure 2.3 u Margin (see Appendix 2.A) u Open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Mechanics of Trading (continued)
n Placing an Offsetting Order u In the exchange-listed options market u In the over-the-counter options market
n Exercising an Option u European vs. American style u Assignment u Cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Option Price Quotations
n See Web sites of newspapers and options exchanges n Problems u Delayed information u Non-synchronized prices
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Types of Options n Stock Options n Index Options n Currency Options n Other Types of Options u interest rate options u currency options u options attached to bonds u exotic options u warrants, callable bonds, convertible bonds u non-traded executive options
n Real Options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Transaction Costs in Option Trading
n Floor Trading and Clearing Fees n Commissions n Bid-Ask Spread n Other Transaction Costs
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
The Regulation of Options Markets
u Federal regulation u Industry regulation u Over-the-counter market regulation u The issue of which agency has regulatory responsibility has occasionally
arisen.
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.A: Margin Requirements
n Definitions u Margin u Initial margin u Maintenance margin
n Margin Requirements on Stock Transactions n Margin Requirements on Option Purchases n Margin Requirements on the Uncovered Sale of Options n Margin Requirements on Covered Calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.B: Taxation of Option Transactions
n Taxation of Long Call Transactions n Taxation of Short Call Transactions n Taxation of Long Put Transactions n Taxation of Short Put Transactions n Taxation of Non-Equity Options n Wash and Constructive Sales
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Types of Options n Stock Options n Index Options n Currency Options n Other Types of Options u interest rate options u currency options u options attached to bonds u exotic options u warrants, callable bonds, convertible bonds u non-traded executive options
n Real Options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Transaction Costs in Option Trading
n Floor Trading and Clearing Fees n Commissions n Bid-Ask Spread n Other Transaction Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
The Regulation of Options Markets
u Federal regulation u Industry regulation u Over-the-counter market regulation u The issue of which agency has regulatory responsibility has occasionally
arisen.
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.A: Margin Requirements
n Definitions u Margin u Initial margin u Maintenance margin
n Margin Requirements on Stock Transactions n Margin Requirements on Option Purchases n Margin Requirements on the Uncovered Sale of Options n Margin Requirements on Covered Calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.B: Taxation of Option Transactions
n Taxation of Long Call Transactions n Taxation of Short Call Transactions n Taxation of Long Put Transactions n Taxation of Short Put Transactions n Taxation of Non-Equity Options n Wash and Constructive Sales
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Types of Options n Stock Options n Index Options n Currency Options n Other Types of Options u interest rate options u currency options u options attached to bonds u exotic options u warrants, callable bonds, convertible bonds u non-traded executive options
n Real Options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Transaction Costs in Option Trading
n Floor Trading and Clearing Fees n Commissions n Bid-Ask Spread n Other Transaction Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
The Regulation of Options Markets
u Federal regulation u Industry regulation u Over-the-counter market regulation u The issue of which agency has regulatory responsibility has occasionally
arisen.
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.A: Margin Requirements
n Definitions u Margin u Initial margin u Maintenance margin
n Margin Requirements on Stock Transactions n Margin Requirements on Option Purchases n Margin Requirements on the Uncovered Sale of Options n Margin Requirements on Covered Calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.B: Taxation of Option Transactions
n Taxation of Long Call Transactions n Taxation of Short Call Transactions n Taxation of Long Put Transactions n Taxation of Short Put Transactions n Taxation of Non-Equity Options n Wash and Constructive Sales
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Types of Options n Stock Options n Index Options n Currency Options n Other Types of Options u interest rate options u currency options u options attached to bonds u exotic options u warrants, callable bonds, convertible bonds u non-traded executive options
n Real Options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Transaction Costs in Option Trading
n Floor Trading and Clearing Fees n Commissions n Bid-Ask Spread n Other Transaction Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
The Regulation of Options Markets
u Federal regulation u Industry regulation u Over-the-counter market regulation u The issue of which agency has regulatory responsibility has occasionally
arisen.
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.A: Margin Requirements
n Definitions u Margin u Initial margin u Maintenance margin
n Margin Requirements on Stock Transactions n Margin Requirements on Option Purchases n Margin Requirements on the Uncovered Sale of Options n Margin Requirements on Covered Calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.B: Taxation of Option Transactions
n Taxation of Long Call Transactions n Taxation of Short Call Transactions n Taxation of Long Put Transactions n Taxation of Short Put Transactions n Taxation of Non-Equity Options n Wash and Constructive Sales
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Types of Options n Stock Options n Index Options n Currency Options n Other Types of Options u interest rate options u currency options u options attached to bonds u exotic options u warrants, callable bonds, convertible bonds u non-traded executive options
n Real Options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Transaction Costs in Option Trading
n Floor Trading and Clearing Fees n Commissions n Bid-Ask Spread n Other Transaction Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
The Regulation of Options Markets
u Federal regulation u Industry regulation u Over-the-counter market regulation u The issue of which agency has regulatory responsibility has occasionally
arisen.
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.A: Margin Requirements
n Definitions u Margin u Initial margin u Maintenance margin
n Margin Requirements on Stock Transactions n Margin Requirements on Option Purchases n Margin Requirements on the Uncovered Sale of Options n Margin Requirements on Covered Calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 2: 0
Appendix 2.B: Taxation of Option Transactions
n Taxation of Long Call Transactions n Taxation of Short Call Transactions n Taxation of Long Put Transactions n Taxation of Short Put Transactions n Taxation of Non-Equity Options n Wash and Constructive Sales
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Chance/Brooks An Introduction to Derivatives and Risk
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Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
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Chance/Brooks An Introduction to Derivatives and Risk
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Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
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Chance/Brooks An Introduction to Derivatives and Risk
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Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Chance/Brooks An Introduction to Derivatives and Risk
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Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 5) (Return to text slide 7)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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(Return to text slide)
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Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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(Return to text slide 5) (Return to text slide 7)
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(Return to text slide)
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Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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(Return to text slide 9)(Return to text slide 8)
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Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
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Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
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Chance/Brooks An Introduction to Derivatives and Risk
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Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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(Return to text slide 5) (Return to text slide 7)
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Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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(Return to text slide 5) (Return to text slide 7)
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Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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(Return to text slide 9)(Return to text slide 8)
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Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
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Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Important Concepts in Chapter 3
n Role of arbitrage in pricing options n Minimum value, maximum value, value at expiration and
lower bound of an option price n Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price n Difference between prices of European and American
options n Put-call parity
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Basic Notation and Terminology
n Symbols u S0 (stock price) u X (exercise price) u T (time to expiration = (days until expiration)/365) u r (Risk Free Rate - see below) u ST (stock price at expiration) u C(S0,T,X), P(S0,T,X)
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Basic Notation and Terminology (continued)
n Computation of risk-free rate (r) u Date: May 14. Option expiration: May 21 u T-bill bid discount = 4.45, ask discount = 4.37 F Average T-bill discount = (4.45+4.37)/2 = 4.41
u T-bill price = 100 - 4.41(7/360) = 99.91425 u T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 u So 4.57 % is risk-free rate for options expiring May 21 u Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
n See Table 3.1 for prices of DCRB options
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Principles of Call Option Pricing n Minimum Value of a Call u C(S0,T,X) ³ 0 (for any call) u For American calls: F Ca(S0,T,X) ³ Max(0,S0 - X)
u Concept of intrinsic value: Max(0,S0 - X) F Proof of intrinsic value rule for DCRB calls
u Concept of time value F See Table 3.2 for time values of DCRB calls
u See Figure 3.1 for minimum values of calls
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(Return to text slide 5) (Return to text slide 7)
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Principles of Call Option Pricing (continued) n Maximum Value of a Call u C(S0,T,X) £ S0 u Intuition u See Figure 3.2, which adds this to Figure 3.1
n Value of a Call at Expiration u C(ST,0,X) = Max(0,ST - X) u Proof/intuition u For American and European options u See Figure 3.3
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(Return to text slide)
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Principles of Call Option Pricing (continued)
n Effect of Time to Expiration u Two American calls differing only by time to expiration, T1 and T2 where
T1 < T2. u Ca(S0,T2,X) ³ Ca(S0,T1,X) F Proof/intuition
u Deep in- and out-of-the-money u Time value maximized when at-the-money u Concept of time value decay u See Figure 3.4 and Table 3.2 u Cannot be proven (yet) for European calls
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(Return to text slide 5) (Return to text slide 7)
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Principles of Call Option Pricing (continued) n Effect of Exercise Price u Effect on Option Value F Two European calls differing only by strikes of X1 and X2. Which is
greater, Ce(S0,T,X1) or Ce(S0,T,X2)? F Construct portfolios A and B. See Table 3.3. F Portfolio A has non-negative payoff; therefore, • Ce(S0,T,X1) ³ Ce(S0,T,X2) • Intuition: show what happens if not true F Prices of DCRB options conform
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(Return to text slide 9)(Return to text slide 8)
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Principles of Call Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.3. We must have • (X2 - X1)(1+r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2) • X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2) • Implications F See Table 3.4. Prices of DCRB options conform
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(Return to text slide 9)(Return to text slide 8)
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Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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(Return to text slide)
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(Return to text slide 15)(Return to text slide 13)
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Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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(Return to text slide)
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Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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(Return to text slide 15)(Return to text slide 13)
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Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide 9)(Return to text slide 8)
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Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide 9)(Return to text slide 8)
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Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 9)(Return to text slide 8)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued) n Lower Bound of a European Call u Construct portfolios A and B. See Table 3.5. u B dominates A. This implies that (after rearranging) F Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] F This is the lower bound for a European call F See Figure 3.5 for the price curve for European calls
u Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´
u For foreign currency calls, F Ce(S0,T,X) ³ Max[0,S0(1+�)-T - X(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n American Call Versus European Call u Ca(S0,T,X) ³ Ce(S0,T,X) u But S0 - X(1+r)-T > S0 - X prior to expiration so F Ca(S0,T,X) ³ Max(0,S0 - X(1+r)-T) F Look at Table 3.6 for lower bounds of DCRB calls
u If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.
F Intuition
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Call Option Pricing (continued)
n Early Exercise of American Calls on Dividend-Paying Stocks
u If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.)
u Intuition
n Effect of Interest Rates n Effect of Stock Volatility
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing n Minimum Value of a Put u P(S0,T,X) ³ 0 (for any put) u For American puts: F Pa(S0,T,X) ³ Max(0,X - S0)
u Concept of intrinsic value: Max(0,X - S0) F Proof of intrinsic value rule for DCRB puts
u See Figure 3.6 for minimum values of puts u Concept of time value F See Table 3.7 for time values of DCRB puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Maximum Value of a Put u Pe(S0,T,X) £ X(1+r)-T u Pa(S0,T,X) £ X u Intuition u See Figure 3.7, which adds this to Figure 3.6
n Value of a Put at Expiration u P(ST,0,X) = Max(0,X - ST) u Proof/intuition u For American and European options u See Figure 3.8
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Time to Expiration u Two American puts differing only by time to expiration, T1 and T2 where
T1 < T2. u Pa(S0,T2,X) ³ Pa(S0,T1,X) F Proof/intuition
u See Figure 3.9 and Table 3.7 u Cannot be proven for European puts
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 15)(Return to text slide 13)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide 16) (Return to text slide 17)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 3: 0
Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
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Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
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Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
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Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
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Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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(Return to text slide 16) (Return to text slide 17)
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Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
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Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
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Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
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Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
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Principles of Put Option Pricing (continued)
n Effect of Exercise Price u Effect on Option Value F Two European puts differing only by X1 and X2. Which is greater,
Pe(S0,T,X1) or Pe(S0,T,X2)? F Construct portfolios A and B. See Table 3.8. F Portfolio A has non-negative payoff; therefore, • Pe(S0,T,X2) ³ Pe(S0,T,X1) • Intuition: show what happens if not true F Prices of DCRB options conform
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(Return to text slide 16) (Return to text slide 17)
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Principles of Put Option Pricing (continued)
n Effect of Exercise Price (continued) u Limits on the Difference in Premiums F Again, note Table 3.8. We must have • (X2 - X1)(1+r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1) • X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1) • Implications F See Table 3.9. Prices of DCRB options conform
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(Return to text slide 16) (Return to text slide 17)
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Principles of Put Option Pricing (continued)
n Lower Bound of a European Put u Construct portfolios A and B. See Table 3.10. u A dominates B. This implies that (after rearranging) F Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) F This is the lower bound for a European put F See Figure 3.10 for the price curve for European puts
u Dividend adjustment: subtract present value of dividends from S to obtain S´
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Principles of Put Option Pricing (continued)
n American Put Versus European Put u Pa(S0,T,X) ³ Pe(S0,T,X)
n Early Exercise of American Puts u There is always a sufficiently low stock price that will make it optimal to
exercise an American put early. u Dividends on the stock reduce the likelihood of early exercise.
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Principles of Put Option Pricing (continued)
n Put-Call Parity u Form portfolios A and B where the options are European. See Table 3.11.
u The portfolios have the same outcomes at the options’ expiration. Thus, it
must be true that F S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T F This is called put-call parity. F It is important to see the alternative ways the equation can be
arranged and their interpretations.
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Principles of Put Option Pricing (continued)
u Put-Call parity for American options can be stated only as inequalities:
u See Table 3.12 for put-call parity for DCRB options u See Figure 3.11 for linkages between underlying asset, risk-free bond, call,
and put through put-call parity.
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Principles of Put Option Pricing (continued)
n The Effect of Interest Rates n The Effect of Stock Volatility
Summary See Table 3.13.
Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise
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Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
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n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
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One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
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n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
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Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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(Return to text slide)
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
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Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
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Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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(Return to text slide)
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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(Return to text slide)
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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(Return to text slide)
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
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Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
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Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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(Return to text slide)
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
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Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Chapter 4: Option Pricing Models: The Binomial Model
Options traders can get by with less math than you think. Tour de France cyclists don't need to know how to solve Newton's laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they're trading.
Emanuel Derman
The Journal of Derivatives, Winter 2000, p. 62
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Important Concepts in Chapter 4
n The concept of an option pricing model n The one- and two-period binomial option pricing models n Explanation of the establishment and maintenance of a
risk-free hedge n Illustration of how early exercise can be captured n The extension of the binomial model to any number of
time periods n Alternative specifications of the binomial model
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n Definition of a model u A simplified representation of reality that uses certain inputs to produce an
output or result
n Definition of an option pricing model u A mathematical formula that uses the factors that determine an option’s
price as inputs to produce the theoretical fair value of an option.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model n Conditions and assumptions u One period, two outcomes (states) u S = current stock price u u = 1 + return if stock goes up u d = 1 + return if stock goes down u r = risk-free rate
n Value of European call at expiration one period later u Cu = Max(0,Su - X) or u Cd = Max(0,Sd - X)
n See Figure 4.1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n Important point: d < 1 + r < u to prevent arbitrage n We construct a hedge portfolio of h shares of stock and one
short call. Current value of portfolio: u V = hS - C
n At expiration the hedge portfolio will be worth u Vu = hSu - Cu u Vd = hSd - Cd u If we are hedged, these must be equal. Setting Vu = Vd and solving for h
gives
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n These values are all known so h is easily computed n Since the portfolio is riskless, it should earn the risk-free
rate. Thus u V(1+r) = Vu (or Vd)
n Substituting for V and Vu u (hS - C)(1+r) = hSu - Cu
n And the theoretical value of the option is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued)
n This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
n The probabilities of the up and down moves were never specified. They are irrelevant to the option price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n An Illustrative Example u S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 u First find the values of Cu, Cd, h, and p: F Cu = Max(0,100(1.25) - 100)
= Max(0,125 - 100) = 25 F Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 F h = (25 - 0)/(125 - 80) = 0.556 F p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6
u Then insert into the formula for C:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
One-Period Binomial Model (continued) n A Hedged Portfolio u Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure
4.2. u Value of investment: V = 556($100) - 1,000($14.02)
$41,580. (This is how much money you must put up.) u Stock goes to $125 F Value of investment = 556($125) - 1,000($25)
= $44,500 u Stock goes to $80 F Value of investment = 556($80) - 1,000($0)
= $44,480
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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One-Period Binomial Model (continued)
n An Overpriced Call u Let the call be selling for $15.00 u Your amount invested is 556($100) - 1,000($15.00)
= $40,600 u You will still end up with $44,500, which is a 9.6% return. u Everyone will take advantage of this, forcing the call price to fall to
$14.02
You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n An Underpriced Call u Let the call be priced at $13 u Sell short 556 shares at $100 and buy 1,000 calls at $13. This will
generate a cash inflow of $42,600. u At expiration, you will end up paying out $44,500. u This is like a loan in which you borrowed $42,600 and paid back $44,500,
a rate of 4.46%, which beats the risk-free borrowing rate.
One-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model
n We now let the stock go up another period so that it ends up Su2, Sud or Sd2.
n See Figure 4.3. n The option expires after two periods with three possible
values:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
n After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values
n The price of the call today will be
Two-Period Binomial Model (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
•The hedge ratios are different in the different states:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n An Illustrative Example u Su2 = 100(1.25)2 = 156.25 u Sud = 100(1.25)(0.80) = 100 u Sd2 = 100(0.80)2 = 64 u The call option prices are as follows
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n The two values of the call at the end of the first period are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n Therefore, the value of the call today is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio u See Figure 4.4. u Call trades at its theoretical value of $17.69. u Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 u So F Buy 701 shares at $100 for $70,100 F Sell 1,000 calls at $17.69 for $17,690 F Net investment: $52,410
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued) n A Hedge Portfolio (continued) u Note each of the possibilities: F Stock goes to 125, then 156.25 F Stock goes to 125, then to 100 F Stock goes to 80, then to 100 F Stock goes to 80, then to 64
u In each case, your wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Hedge Portfolio (continued) u Your wealth then grows by 7% from the end of the first period to the end
of the second. u Conclusion: If the option is correctly priced and you maintain the
appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Two-Period Binomial Model (continued)
n A Mispriced Call in the Two-Period World u If the call is underpriced, you buy it and short the stock, maintaining the
correct hedge over both periods. You end up borrowing at less than the risk-free rate.
u If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.
u See Table 4.1 for summary.
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model
n Pricing Put Options u Same procedure as calls but use put payoff formula at expiration. In our
example the put prices at expiration are
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Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u The two values of the put at the end of the first period are
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Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Therefore, the value of the put today is
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Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Let us hedge a long position in stock by purchasing puts. The hedge ratio
formula is the same except that we ignore the negative sign:
u Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.
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Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 125. We now have F 299 shares at $125 + 1,000 puts at $0.0 = $37,375 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.
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Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 100 to 80. We now have F 299 shares at $80 + 1,000 puts at $13.46 = $37,380 F This is a 7% gain over $34,930. The new hedge ratio is
u Thus, buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.
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Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 125 to 156.25. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
u Stock goes from 125 to 100. We now have F Bond worth $37,375(1.07) = $39,991 F This is a 7% gain.
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Extensions of the Binomial Model (continued)
n Pricing Put Options (continued) u Stock goes from 80 to 100. We now have F 1,000 shares worth $100 each, 1,000 puts worth $0 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
u Stock goes from 80 to 64. We now have F 1,000 shares worth $64 each, 1,000 puts worth $36 each, plus a loan
in which we owe $56,080(1.07) = $60,006 for a total of $39,994, a 7% gain
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n American Puts and Early Exercise u Now we must consider the possibility of exercising the put early. At time
1 the European put values were F Pu = 0.00 when the stock is at 125 F Pd = 13.46 when the stock is at 80
u When the stock is at 80, the put is in-the-money by $20 so exercise it early. Replace Pu = 13.46 with Pu = 20. The value of the put today is higher at
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise u One way to incorporate dividends is to assume a constant yield, �, per
period. The stock moves up, then drops by the rate �. u See Figure 4.5 for example with a 10% yield u The call prices at expiration are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The European call prices after one period are
u The European call value at time 0 is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u If the call is American, when the stock is at 125, it pays a dividend of
$12.50 and then falls to $112.50. We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex-dividend, falling to $112.50 but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of Cu = $22.78 with Cu = $25. At time 0 the value is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Alternatively, we can specify that the stock pays a specific dollar dividend
at time 1. Assume $12. Unfortunately, the tree no longer recombines, as in Figure 4.6. We can still calculate the option value but the tree grows large very fast. See Figure 4.7.
u Because of the reduction in the number of computations, trees that recombine are preferred over trees that do not recombine.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u Yet another alternative (and preferred) specification is to subtract the
present value of the dividends from the stock price (as we did in Chapter 3) and let the adjusted stock price follow the binomial up and down factors. For this problem, see Figure 4.8.
u The tree now recombines and we can easily calculate the option values following the same procedure as before.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u The option prices at expiration are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Dividends, European Calls, American Calls, and Early
Exercise (continued) u At time 1 the option prices are
u We exercise at time 1 so that Cu is now 22.99. At time 0
u The European option value would be 12.18.
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Foreign Currency Options u Underlying instrument is currency u Holding of foreign currency can earn the foreign risk-free interest rate u The binomial probability is altered to adjust for the foreign risk-free interest
rate effect u The binomial probability is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n Extending the Binomial Model to n Periods u With n periods to go, the binomial model can be easily extended. There is
a long and somewhat complex looking formula in the book. The basic procedure, however, is the same. See Figure 4.9 in which we see below the stock prices the prices of European and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods.
u At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n Behavior of the Binomial Model for Large n and a Fixed
Option Life u The risk-free rate is adjusted to (1 + r)T/n-1 u The up and down parameters are adjusted to
F where � is the volatility. Let us price the DCRB June 125 call with
one period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) u The parameters are now
u The new stock prices are F Su = 125.9375(1.293087) = 162.8481 F Sd = 125.9375(0.773343) = 97.3929
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Chance/Brooks An Introduction to Derivatives and Risk
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Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The new option prices would be • Cu = Max(0,162.8481-125) = 37.85 • Cd = Max(0,97.3929 - 125) = 0.0 F p would be (1.004285 - 0.773343)/(1.293087 - 0.773343) = 0.444; 1 -
p = 0.556. F The price of the option at time 0 is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The actual price of the option is 13.50, but obviously one binomial
period is not enough. F Table 4.2 shows what happens as we increase the number of binomial
periods. The price converges to around 13.56. In Chapter 5, we shall see that this is approximately the Black-Scholes-Merton price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model u We can use a different specification of u, d and p
u where ln(1 + r) is the continuously compounded interest rate. Here p will converge to 0.5 as n increases.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model (continued)
u Now let us price the DCRB June 125 call but use two periods. We have r = (1.0456)0.0959/2 - 1 = 0.0021. Using our previous formulas,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model n Alternative Specifications of the Binomial Model (continued) u Now let us use these new formulas:
u We can use 0.5 for p. See Figure 4.10. The prices are close and will converge when n is large.
u See BSMbin9e.xls for software to calculate the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The new option prices would be • Cu = Max(0,162.8481-125) = 37.85 • Cd = Max(0,97.3929 - 125) = 0.0 F p would be (1.004285 - 0.773343)/(1.293087 - 0.773343) = 0.444; 1 -
p = 0.556. F The price of the option at time 0 is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The actual price of the option is 13.50, but obviously one binomial
period is not enough. F Table 4.2 shows what happens as we increase the number of binomial
periods. The price converges to around 13.56. In Chapter 5, we shall see that this is approximately the Black-Scholes-Merton price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model u We can use a different specification of u, d and p
u where ln(1 + r) is the continuously compounded interest rate. Here p will converge to 0.5 as n increases.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model (continued)
u Now let us price the DCRB June 125 call but use two periods. We have r = (1.0456)0.0959/2 - 1 = 0.0021. Using our previous formulas,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model n Alternative Specifications of the Binomial Model (continued) u Now let us use these new formulas:
u We can use 0.5 for p. See Figure 4.10. The prices are close and will converge when n is large.
u See BSMbin9e.xls for software to calculate the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The new option prices would be • Cu = Max(0,162.8481-125) = 37.85 • Cd = Max(0,97.3929 - 125) = 0.0 F p would be (1.004285 - 0.773343)/(1.293087 - 0.773343) = 0.444; 1 -
p = 0.556. F The price of the option at time 0 is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The actual price of the option is 13.50, but obviously one binomial
period is not enough. F Table 4.2 shows what happens as we increase the number of binomial
periods. The price converges to around 13.56. In Chapter 5, we shall see that this is approximately the Black-Scholes-Merton price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model u We can use a different specification of u, d and p
u where ln(1 + r) is the continuously compounded interest rate. Here p will converge to 0.5 as n increases.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model (continued)
u Now let us price the DCRB June 125 call but use two periods. We have r = (1.0456)0.0959/2 - 1 = 0.0021. Using our previous formulas,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model n Alternative Specifications of the Binomial Model (continued) u Now let us use these new formulas:
u We can use 0.5 for p. See Figure 4.10. The prices are close and will converge when n is large.
u See BSMbin9e.xls for software to calculate the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The new option prices would be • Cu = Max(0,162.8481-125) = 37.85 • Cd = Max(0,97.3929 - 125) = 0.0 F p would be (1.004285 - 0.773343)/(1.293087 - 0.773343) = 0.444; 1 -
p = 0.556. F The price of the option at time 0 is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The actual price of the option is 13.50, but obviously one binomial
period is not enough. F Table 4.2 shows what happens as we increase the number of binomial
periods. The price converges to around 13.56. In Chapter 5, we shall see that this is approximately the Black-Scholes-Merton price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model u We can use a different specification of u, d and p
u where ln(1 + r) is the continuously compounded interest rate. Here p will converge to 0.5 as n increases.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model (continued)
u Now let us price the DCRB June 125 call but use two periods. We have r = (1.0456)0.0959/2 - 1 = 0.0021. Using our previous formulas,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model n Alternative Specifications of the Binomial Model (continued) u Now let us use these new formulas:
u We can use 0.5 for p. See Figure 4.10. The prices are close and will converge when n is large.
u See BSMbin9e.xls for software to calculate the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The new option prices would be • Cu = Max(0,162.8481-125) = 37.85 • Cd = Max(0,97.3929 - 125) = 0.0 F p would be (1.004285 - 0.773343)/(1.293087 - 0.773343) = 0.444; 1 -
p = 0.556. F The price of the option at time 0 is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The actual price of the option is 13.50, but obviously one binomial
period is not enough. F Table 4.2 shows what happens as we increase the number of binomial
periods. The price converges to around 13.56. In Chapter 5, we shall see that this is approximately the Black-Scholes-Merton price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model u We can use a different specification of u, d and p
u where ln(1 + r) is the continuously compounded interest rate. Here p will converge to 0.5 as n increases.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model (continued)
u Now let us price the DCRB June 125 call but use two periods. We have r = (1.0456)0.0959/2 - 1 = 0.0021. Using our previous formulas,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model n Alternative Specifications of the Binomial Model (continued) u Now let us use these new formulas:
u We can use 0.5 for p. See Figure 4.10. The prices are close and will converge when n is large.
u See BSMbin9e.xls for software to calculate the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The new option prices would be • Cu = Max(0,162.8481-125) = 37.85 • Cd = Max(0,97.3929 - 125) = 0.0 F p would be (1.004285 - 0.773343)/(1.293087 - 0.773343) = 0.444; 1 -
p = 0.556. F The price of the option at time 0 is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The actual price of the option is 13.50, but obviously one binomial
period is not enough. F Table 4.2 shows what happens as we increase the number of binomial
periods. The price converges to around 13.56. In Chapter 5, we shall see that this is approximately the Black-Scholes-Merton price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model u We can use a different specification of u, d and p
u where ln(1 + r) is the continuously compounded interest rate. Here p will converge to 0.5 as n increases.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model (continued)
u Now let us price the DCRB June 125 call but use two periods. We have r = (1.0456)0.0959/2 - 1 = 0.0021. Using our previous formulas,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model n Alternative Specifications of the Binomial Model (continued) u Now let us use these new formulas:
u We can use 0.5 for p. See Figure 4.10. The prices are close and will converge when n is large.
u See BSMbin9e.xls for software to calculate the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The new option prices would be • Cu = Max(0,162.8481-125) = 37.85 • Cd = Max(0,97.3929 - 125) = 0.0 F p would be (1.004285 - 0.773343)/(1.293087 - 0.773343) = 0.444; 1 -
p = 0.556. F The price of the option at time 0 is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The actual price of the option is 13.50, but obviously one binomial
period is not enough. F Table 4.2 shows what happens as we increase the number of binomial
periods. The price converges to around 13.56. In Chapter 5, we shall see that this is approximately the Black-Scholes-Merton price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model u We can use a different specification of u, d and p
u where ln(1 + r) is the continuously compounded interest rate. Here p will converge to 0.5 as n increases.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model (continued)
u Now let us price the DCRB June 125 call but use two periods. We have r = (1.0456)0.0959/2 - 1 = 0.0021. Using our previous formulas,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model n Alternative Specifications of the Binomial Model (continued) u Now let us use these new formulas:
u We can use 0.5 for p. See Figure 4.10. The prices are close and will converge when n is large.
u See BSMbin9e.xls for software to calculate the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The new option prices would be • Cu = Max(0,162.8481-125) = 37.85 • Cd = Max(0,97.3929 - 125) = 0.0 F p would be (1.004285 - 0.773343)/(1.293087 - 0.773343) = 0.444; 1 -
p = 0.556. F The price of the option at time 0 is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model (continued) n The Behavior of the Binomial Model for Large n and a
Fixed Option Life (continued) F The actual price of the option is 13.50, but obviously one binomial
period is not enough. F Table 4.2 shows what happens as we increase the number of binomial
periods. The price converges to around 13.56. In Chapter 5, we shall see that this is approximately the Black-Scholes-Merton price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model u We can use a different specification of u, d and p
u where ln(1 + r) is the continuously compounded interest rate. Here p will converge to 0.5 as n increases.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 4: 0
Extensions of the Binomial Model
n Alternative Specifications of the Binomial Model (continued)
u Now let us price the DCRB June 125 call but use two periods. We have r = (1.0456)0.0959/2 - 1 = 0.0021. Using our previous formulas,
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Extensions of the Binomial Model n Alternative Specifications of the Binomial Model (continued) u Now let us use these new formulas:
u We can use 0.5 for p. See Figure 4.10. The prices are close and will converge when n is large.
u See BSMbin9e.xls for software to calculate the binomial model.
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(Return to text slide)
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Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Chapter 5: Option Pricing Models: The Black-Scholes-Merton Model
Good theories, like Black-Scholes-Merton, provide a theoretical laboratory in which you can explore the likely effect of possible causes. They give you a common language with which to quantify and communicate your feelings about value.
Emanuel Derman The Journal of Derivatives, Winter, 2000, p. 64
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Important Concepts in Chapter 5
n The Black-Scholes-Merton option pricing model n The relationship of the model’s inputs to the option price n How to adjust the model to accommodate dividends and
put options n The concepts of historical and implied volatility n Hedging an option position
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Origins of the Black-Scholes-Merton Formula
n Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
n Black, Scholes, Merton and the 1997 Nobel Prize
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Black-Scholes-Merton Model as the Limit of the Binomial Model
n Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
n Consider the DCRB June 125 call option. Figure 5.1 shows the model price for an increasing number of time steps.
n The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.
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Assumptions of the Model n Stock prices behave randomly and evolve according to a
lognormal distribution. u See Figure 5.2a, 5.2b and 5.3 for a look at the notion of randomness. u A lognormal distribution means that the log (continuously compounded)
return is normally distributed. See Figure 5.4. n The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life n There are no taxes or transaction costs n The stock pays no dividends n The options are European
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A Nobel Formula
n The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
n The model is derived with complex mathematics but is easily understandable. The formula is
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A Nobel Formula (continued)
u where F N(d1), N(d2) = cumulative normal probability F s = annualized standard deviation (volatility) of the continuously
compounded return on the stock F rc = continuously compounded risk-free rate
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A Nobel Formula (continued)
n A Digression on Using the Normal Distribution u The familiar normal, bell-shaped curve
(Figure 5.5) u See Table 5.1 for determining the normal probability for d1 and d2. This
gives you N(d1) and N(d2).
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A Nobel Formula (continued)
n A Numerical Example u Price the DCRB June 125 call u S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959, s = 0.83. u See Table 5.2 for calculations. C = $13.21. u Familiarize yourself with the accompanying software F BSMbin8e.xls. Note the use of Excel’s =normsdist() function. F BSMImpVol8e.xls. See Appendix.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
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Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula u Interpretation of the Formula F The concept of risk neutrality, risk neutral probability, and its role in
pricing options F The option price is the discounted expected payoff, Max(0,ST - X).
We need the expected value of ST - X for those cases where ST > X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price
given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present.
F The second term is the expected value of the payment of the exercise price at expiration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Black-Scholes-Merton Formula and the Lower Bound of a European
Call F Recall from Chapter 3 that the lower bound would be
F The Black-Scholes-Merton formula always exceeds this value as seen
by letting S0 be very high and then let it approach zero.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When T = 0 F At expiration, the formula must converge to the intrinsic value. F It does but requires taking limits since otherwise it would be division
by zero. F Must consider the separate cases of ST � X and
ST < X.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When S0 = 0 F Here the company is bankrupt so the formula must converge to zero. F It requires taking the log of zero, but by taking limits we obtain the
correct result.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When � = 0 F Again, this requires dividing by zero, but we can take limits and
obtain the right answer F If the option is in-the-money as defined by the stock price exceeding
the present value of the exercise price, the formula converges to the stock price minus the present value of the exercise price. Otherwise, it converges to zero.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When X = 0 F From Chapter 3, the call price should converge to the stock price. F Here both N(d1) and N(d2) approach 1.0 so by taking limits, the
formula converges to S0.
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A Nobel Formula (continued) n Characteristics of the Black-Scholes-Merton Formula
(continued) u The Formula When rc = 0 F A zero interest rate is not a special case and no special result is
obtained.
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Variables in the Black-Scholes-Merton Model n The Stock Price u Let S , then C . See Figure 5.6. u This effect is called the delta, which is given by N(d1). u Measures the change in call price over the change in stock price for a very
small change in the stock price. u Delta ranges from zero to one. See Figure 5.7 for how delta varies with
the stock price. u The delta changes throughout the option’s life. See Figure 5.8.
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging/delta neutral: holding shares of stock and selling calls to
maintain a risk-free position F The number of shares held per option sold is the delta, N(d1). F As the stock goes up/down by $1, the option goes up/down by N(d1).
By holding N(d1) shares per call, the effects offset. F The position must be adjusted as the delta changes.
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u Delta hedging works only for small stock price changes. For larger
changes, the delta does not accurately reflect the option price change. This risk is captured by the gamma:
u For our DCRB June 125 call,
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Variables in the Black-Scholes-Merton Model (continued)
n The Stock Price (continued) u If the stock goes from 125.94 to 130, the delta is predicted to change from
0.569 to 0.569 + (130 - 125.94)(0.0123) = 0.6189. The actual delta at a price of 130 is 0.6171. So gamma captures most of the change in delta.
u The larger is the gamma, the more sensitive is the option price to large stock price moves, the more sensitive is the delta, and the faster the delta changes. This makes it more difficult to hedge.
u See Figure 5.9 for gamma vs. the stock price u See Figure 5.10 for gamma vs. time
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Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
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Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
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Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
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Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
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Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
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Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
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Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
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n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
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Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
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Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
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Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
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Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
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Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
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Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
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Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
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Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
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n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
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Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
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Chance/Brooks An Introduction to Derivatives and Risk
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Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
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Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
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(Return to text slide)
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Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
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(Return to text slide)
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Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Exercise Price u Let X , then C ¯ u The exercise price does not change in most options so this is useful only
for comparing options differing only by a small change in the exercise price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Risk-Free Rate u Take ln(1 + discrete risk-free rate from Chapter 3). u Let rc , then C . See Figure 5.11. The effect is called rho
u In our example,
u If the risk-free rate goes to 0.12, the rho estimates that the call price will
go to (0.12 - 0.0446)(5.57) = 0.42. The actual change is 0.43. u See Figure 5.12 for rho vs. stock price.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation u The most critical variable in the Black-Scholes-Merton model because the
option price is very sensitive to the volatility and it is the only unobservable variable.
u Let s , then C . See Figure 5.13. u This effect is known as vega.
u In our problem this is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Volatility or Standard Deviation (continued) u Thus if volatility changes by 0.01, the call price is estimated to change by
15.32(0.01) = 0.15 u If we increase volatility to, say, 0.95, the estimated change would be
15.32(0.12) = 1.84. The actual call price at a volatility of 0.95 would be 15.39, which is an increase of 1.84. The accuracy is due to the near linearity of the call price with respect to the volatility.
u See Figure 5.14 for the vega vs. the stock price. Notice how it is highest when the call is approximately at-the-money.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration u Calculated as (days to expiration)/365 u Let T , then C . See Figure 5.15. This effect is known as theta:
n In our problem, this would be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Variables in the Black-Scholes-Merton Model (continued)
n The Time to Expiration (continued) u If one week elapsed, the call price would be expected to change to (0.0959
- 0.0767)(-68.91) = -1.32. The actual call price with T = 0.0767 is 12.16, a decrease of 1.39.
u See Figure 5.16 for theta vs. the stock price u Note that your spreadsheet BSMbin8e.xls calculate the delta, gamma,
vega, theta, and rho for calls and puts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Black-Scholes-Merton Model When the Stock Pays Dividends
n Known Discrete Dividends u Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life. u Subtract present value of dividends from stock price. u Adjusted stock price, S¢, is inserted into the B-S-M model:
u See Table 5.3 for example. u The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
n Continuous Dividend Yield u Assume the stock pays dividends continuously at the rate of �. u Subtract present value of dividends from stock price. Adjusted
stock price, S¢, is inserted into the B-S model.
u See Table 5.4 for example. u This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously compounded foreign risk-free rate.
u The Excel spreadsheet BSMbin8e.xls permit you to enter a continuous dividend yield.
Black-Scholes-Merton Model When the Stock Pays Dividends (continued)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Black-Scholes-Merton Model and Some Insights into American Call Options
n Table 5.5 illustrates how the early exercise decision is made when the dividend is the only one during the option’s life
n The value obtained upon exercise is compared to the ex- dividend value of the option.
n High dividends and low time value lead to early exercise. n Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
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(Return to text slide)
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Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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(Return to text slide)
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(Return to text slide)
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Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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(Return to text slide)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 5: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To continue) (Return to text slide 38)
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(Return to text slide)
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Management, 9th ed. Ch. 5: 0
Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
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Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
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Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
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(To continue) (Return to text slide 38)
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Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
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Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
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(Return to text slide)
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Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
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Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
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(To continue) (Return to text slide 38)
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Estimating the Volatility
n Historical Volatility u This is the volatility over a recent time period. u Collect daily, weekly, or monthly returns on the stock. u Convert each return to its continuously compounded equivalent by taking
ln(1 + return). Calculate variance. u Annualize by multiplying by 250 (daily returns), 52 (weekly returns) or 12
(monthly returns). Take square root. See Table 5.6 for example with DCRB.
u Your Excel spreadsheet Hisv8e.xls will do these calculations. See Software Demonstration 5.2.
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Estimating the Volatility (continued) n Implied Volatility u This is the volatility implied when the market price of the option is set to
the model price. u Figure 5.17 illustrates the procedure. u Substitute estimates of the volatility into the B-S-M formula until the
market price converges to the model price. See Table 5.7 for the implied volatilities of the DCRB calls.
u A short-cut for at-the-money options is
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(Return to text slide)
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Estimating the Volatility (continued) n Implied Volatility (continued) u For our DCRB June 125 call, this gives
u This is quite close; the actual implied volatility is 0.83. u Appendix 5.A shows a method to produce faster convergence.
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Estimating the Volatility (continued) n Implied Volatility (continued) u Interpreting the Implied Volatility F The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See Figure 5.18.
F The relationship between the implied volatility and the exercise price is called the volatility smile or volatility skew. Figure 5.19. These volatilities are actually supposed to be the same. This effect is puzzling and has not been adequately explained.
F The CBOE has constructed indices of implied volatility of one-month at-the-money options based on the S&P 100 (VIX) and Nasdaq (VXN). See Figure 5.20.
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Put Option Pricing Models n Restate put-call parity with continuous discounting
n Substituting the B-S-M formula for C above gives the B-S-M put option pricing model
n N(d1) and N(d2) are the same as in the call model.
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Put Option Pricing Models (continued) n Note calculation of put price:
n The Black-Scholes-Merton price does not reflect early exercise and, thus, is extremely biased here since the American option price in the market is 11.50. A binomial model would be necessary to get an accurate price. With n = 100, we obtained 12.11.
n See Table 5.8 for the effect of the input variables on the Black- Scholes-Merton put formula.
n Your software also calculates put prices and Greeks.
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Managing the Risk of Options n Here we talk about how option dealers hedge the risk of
option positions they take. n Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of 0.5692. Dealer will buy 569 shares and adjust the hedge daily.
u To buy 569 shares at $125.94 and sell 1,000 calls at $13.5533 will require $58,107.
u We simulate the daily stock prices for 35 days, at which time the call expires.
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Managing the Risk of Options (continued) n The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price of 10.4078 and delta of 0.4981. We have
u Stock worth 569($120.4020) = $68,509 u Options worth -1,000($10.4078) = -$10,408 u Total of $58,101 u Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114. n Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71 and invest the money ($8,549) in bonds.
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(To continue) (Return to text slide 38)
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(To previous slide) (Return to text slide 38)
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Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
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Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
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Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
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Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
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Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
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When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
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Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
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(Return to text slide)
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Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
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Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
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Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
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(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(To previous slide) (Return to text slide 38)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have u Stock worth 498($126.2305) = $62,863 u Options worth -1,000($13.3358) = -$13,336 u Bonds worth $8,550 (includes one days’ interest) u Total of $58,077 u Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44. n At the end we have $59,762, a excess of $1,406.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade continuously result in imperfections in the delta hedge.
n To deal with this problem, we must gamma hedge, i.e., reduce the gamma to zero. We can do this only by adding another option. Let us use the June 130 call, selling at 11.3792 with a delta of 0.5087 and gamma of 0.0123. Our original June 125 call has a gamma of 0.0121. The stock gamma is zero.
n We shall use the symbols �1, �2, �1 and �2. We use hS shares of stock and hC of the June 130 calls.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The delta hedge condition is u hS(1) - 1,000�1 + hC � 2 = 0
n The gamma hedge condition is u -1,000�1 + hC �2 = 0
n We can solve the second equation and get hC and then substitute back into the first to get hS. Solving for hC and hS, we obtain
u hC = 1,000(0.0121/0.0123) = 984 u hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
n So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The initial outlay will be u 68($125.94) - 1,000($13.5533) + 985($11.3792) = $6,219
n At the end of day one, the stock is at 120.4020, the 125 call is at 10.4078, the 130 call is at 8.5729. The portfolio is worth
u 68($120.4020) - 1,000($10.4078) + 985($8.5729) = $6,224
n It should be worth $6,218e0.0446(1/365) = $6,220. n The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Managing the Risk of Options (continued) n The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall, this generates $1,444, which we invest in bonds.
n The next day, the stock is at $126.2305, the 125 call is at $13.3358 and the 130 call is at $11.1394. The bonds are worth $1,205. The portfolio is worth
u 56($126.2305) - 1,000($13.3358) + 1,013($11.1394) + $1,205 = $6,222. n The portfolio should be worth $6,219e0.0446(2/365) =
$6,221. n Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than when only delta hedging.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
When the Black-Scholes-Merton may or may not hold
n Liquidity n Short-Selling n Information Asymmetry n Problems with Exotic Options n Performativity and Counter-Performativity
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Summary n See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and Black- Scholes-Merton call and put option pricing models.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility
n This technique developed by Manaster and Koehler gives a starting point and guarantees convergence. Let a given volatility be �* and the corresponding Black-Scholes- Merton price be C(�*). The initial guess should be
n You then compute C(�1*). If it is not close enough, you make the next guess.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n Given the ith guess, the next guess should be
n where d1 is computed using �1*. Let us illustrate using the DCRB June 125 call. C(�) = 13.50. The initial guess is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 5: 0
Appendix 5.A: A Shortcut to the Calculation of Implied Volatility (continued)
n At a volatility of 0.4950, the Black-Scholes-Merton value is 8.41. The next guess should be
n where 0.1533 is d1 computed from the Black-Scholes- Merton-Merton model using 0.4950 as the volatility and 2.5066 is the square root of 2�. Now using 0.8260, we obtain a Black-Scholes-Merton value of 13.49, which is close enough to 13.50. So 0.83 is the implied volatility.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
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Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
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Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
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Management, 9th ed. Ch. 6: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Chance/Brooks An Introduction to Derivatives and Risk
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Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
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Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
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Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
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Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
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Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
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Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
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Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
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Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
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Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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(Return to text slide)
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Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
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(Return to text slide)
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Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
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Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Important Concepts in Chapter 6
n Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals
n The effect of choosing different exercise prices n The effect of closing out an option position early versus
holding to expiration
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Chance/Brooks An Introduction to Derivatives and Risk
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Terminology and Notation n Note the following standard symbols u C = current call price, P = current put price u S0 = current stock price, ST = stock price at expiration u T = time to expiration u X = exercise price u P = profit from strategy
n The number of calls, puts and stock is given as u NC = number of calls u NP = number of puts u NS = number of shares of stock
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Terminology and Notation (continued) n These symbols imply the following: u NC, NP, or NS > 0 implies buying (going long) u NC, NP, or NS < 0 implies selling (going short)
n The Profit Equations u Profit equation for calls held to expiration F P = NC[Max(0,ST - X) - C] • For buyer of one call (NC = 1) this implies P = Max(0,ST -
X) - C • For seller of one call (NC = -1) this implies P =
-Max(0,ST - X) + C
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Terminology and Notation (continued) n The Profit Equations (continued) u Profit equation for puts held to expiration F P = NP[Max(0,X - ST) - P] • For buyer of one put (NP = 1) this implies P = Max(0,X -
ST) - P • For seller of one put (NP = -1) this implies P = -Max(0,X -
ST) + P
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Terminology and Notation (continued)
n The Profit Equations (continued) u Profit equation for stock F P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P = ST - S0 • For short seller of one share (NS = -1) this implies P = -ST + S0
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Terminology and Notation (continued) n Different Holding Periods u Three holding periods: T1 < T2 < T u For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.
F Remaining time to expiration will be either T - T1, T - T2 or T - T = 0 (we have already covered the latter)
F For a position closed out at T1, the profit will be
F where the closeout option price is taken from the Black- Scholes-Merton model for a given stock price at T1.
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Terminology and Notation (continued)
n Different Holding Periods (continued) u Similar calculation done for T2 u For T, the profit is determined by the intrinsic value, as already covered
n Assumptions u No dividends u No taxes or transaction costs u We continue with the DCRB options. See
Table 6.1.
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Stock Transactions
n Buy Stock u Profit equation: P = NS[ST - S0] given that NS > 0 u See Figure 6.1 for DCRB, S0 = $125.94 u Maximum profit = �, minimum = -S0
n Sell Short Stock u Profit equation: P = NS[ST - S0] given that NS < 0 u See Figure 6.2 for DCRB, S0 = $125.94 u Maximum profit = S0, minimum = - �
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(Return to text slide)
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Call Option Transactions
n Buy a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting
NC = 1, F P = ST - X - C if ST > X F P = - C if ST £ X
u See Figure 6.3 for DCRB June 125, C = $13.50 u Maximum profit = �, minimum = -C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X + C
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Call Option Transactions (continued)
n Buy a Call (continued) u See Figure 6.4 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.
u Note how time value decay affects profit for given holding period.
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(Return to text slide)
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Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
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Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
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Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
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Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
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Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
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Management, 9th ed. Ch. 6: 0
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Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
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Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
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Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call u Profit equation: P = NC[Max(0,ST - X) - C] given that NC < 0. Letting
NC = -1, F P = -ST + X + C if ST > X F P = C if ST £ X
u See Figure 6.6 for DCRB June 125, C = $13.50 u Maximum profit = +C, minimum = - � u Breakeven stock price same as buying call: ST* = X + C
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Call Option Transactions (continued)
n Write a Call (continued) u See Figure 6.7 for different exercise prices. Note differences in maximum
loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.8. u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions
n Buy a Put u Profit equation: P = NP[Max(0,X - ST) - P] given that NP > 0. Letting
NP = 1, F P = X - ST - P if ST < X F P = - P if ST ³ X
u See Figure 6.9 for DCRB June 125, P = $11.50 u Maximum profit = X - P, minimum = -P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Buy a Put (continued) u See Figure 6.10 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.11. u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put u Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP
= -1 F P = -X + ST + P if ST < X F P = P if ST ³ X
u See Figure 6.12 for DCRB June 125, P = $11.50 u Maximum profit = +P, minimum = -X + P u Breakeven stock price found by setting profit equation to zero and solving:
ST* = X - P
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Put Option Transactions (continued)
n Write a Put (continued) u See Figure 6.13 for different exercise prices. Note differences in
maximum loss and breakeven. u For different holding periods, compute profit for range of stock prices at
T1, T2, and T using Black-Scholes-Merton model. See Figure 6.14. u Note how time value decay affects profit for given holding period.
n Figure 6.15 summarizes these payoff graphs.
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Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
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Chance/Brooks An Introduction to Derivatives and Risk
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Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
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Chance/Brooks An Introduction to Derivatives and Risk
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Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call
u One short call for every share owned u Profit equation: P = NS(ST - S0) + NC[Max(0,ST - X) - C] given NS > 0, NC
< 0, NS = -NC. With NS = 1, NC = -1, F P = ST - S0 + C if ST £ X F P = X - S0 + C if ST > X
u See Figure 6.16 for DCRB June 125, S0 = $125.94, C = $13.50
u Maximum profit = X - S0 + C, minimum = -S0 + C u Breakeven stock price found by setting profit equation to zero and solving:
ST* = S0 - C
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Calls and Stock: the Covered Call (continued)
u See Figure 6.17 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.18.
u Note the effect of time value decay. u Some General Considerations for Covered Calls: F alleged attractiveness of the strategy F misconception about picking up income F rolling up to avoid exercise
u Opposite is short stock, buy call
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Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put u One long put for every share owned u Profit equation: P = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1, F P = ST - S0 - P if ST ³ X F P = X - S0 - P if ST < X
u See Figure 6.19 for DCRB June 125, S0 = $125.94, P = $11.50
u Maximum profit = �, minimum = X - S0 - P u Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0 u Like insurance policy
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Puts and Stock: the Protective Put (continued)
u See Figure 6.20 for different exercise prices. Note differences in maximum loss and breakeven.
u For different holding periods, compute profit for range of stock prices at T1, T2, and T using Black-Scholes-Merton model. See Figure 6.21.
u Note how time value decay affects profit for given holding period.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls u Rearranging put-call parity to isolate put price
u This implies put = long call, short stock, long risk-free bond with face value X.
u This is a synthetic put. u In practice most synthetic puts are constructed without risk-free bond, i.e.,
long call, short stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
Synthetic Puts and Calls (continued) u Profit equation: P = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP. Letting NC = 1, NS = -1,
F P = -C - ST + S0 if ST £ X F P = S0 - X - C if ST > X
u See Figure 6.22 for synthetic put vs. actual put. u Table 6.3 shows payoffs from reverse conversion (long call, short stock,
short put), used when actual put is overpriced. Like risk-free borrowing. u Similar strategy for conversion, used when actual call overpriced.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 6: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chapter 8: The Structure of Forward and Futures Markets
Futures traders tend to be superstitious—when on a good run they are reluctant to change their mojo, this includes washing their jackets. Traders will wear their lucky jackets until they fall apart or their luck runs out. Some traders have even been buried in their lucky jackets, reflecting a hope that the good luck their jackets provided in the trading pits on Earth could be retained for eternity in that Great Trading Pit in the sky.
Jim Overdahl Futures Fall Special Issue 2005, p. 14
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Important Concepts in Chapter 8 n Definitions and examples of forward and futures contracts n Institutional characteristics of forward and futures markets n Futures contracts available for trading n Placing an order, margins, daily settlement n The role of the clearinghouse n Accessing futures price quotations n Magnitude and effects of transaction costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Development of Forward and Futures Markets
n Chicago Futures Markets n Development of Financial Futures n Development of Options on Futures Markets n Parallel Development of Over-the-Counter Markets u interbank market u growth of forward markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Over-the-Counter Forward Market
u customized u private u essentially unregulated u credit risk u market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010 u See Figure 8.1 for notional amount of forward market u See Figure 8.2 for market value of forward market
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading n Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract) n Contract Terms and Conditions u contract size u quotation unit u minimum price fluctuation u contract grade u trading hours
n Delivery Terms u delivery date and time u delivery or cash settlement
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Organized Futures Trading (continued)
n Daily Price Limits and Trading Halts u limit moves u circuit breakers
n Other Exchange Responsibilities u minimum financial responsibility requirements u position limits u rules governing the trading floor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Derivatives Exchanges
n Global and after-hours trading n Estimated world-wide volume in 2010 was 11.2 billion
contracts n 43% Asia Pacific Region n 13% North America n 3.7 billion at Korea Exchange n 3.1 billion at CME Group
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders
n General Classes of Futures Traders u futures commission merchants u locals u dual trading
n Classification by Trading Strategy u hedger/speculator u spreader u arbitrageur
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Traders (continued) n Classification by Trading Style u scalpers u day traders u position traders
n Off-Floor Futures Traders u individuals u institutions u Others: Introducing Broker (IB), Commodity Trading Advisor (CTA),
Commodity Pool Operator (CPO), Associated Person (AP)
n Forward Market Traders u over-the-counter u primarily institutions
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading
n Placing an Order u pit u open outcry u electronic systems
n Role of the Clearinghouse u See Figure 8.4. u margin deposits
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Daily Settlement u initial margin u maintenance margin u concept of “margin” vs. performance bond u settlement price u variation margin u See Table 8.1 for example. u open interest
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Mechanics of Futures Trading (continued)
n Delivery and Cash Settlement u three-day delivery process u alternative deliverable grades u offsetting u exchange for physicals u forward market procedures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Price Quotations
n Newspapers (such as The Wall Street Journal)
n Web sites
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Types of Futures Contracts n Agricultural Commodities n Natural Resources n Miscellaneous Commodities n Foreign Currencies n Federal funds and Eurodollars n Treasury Notes and Bonds n Swap Futures n Equities n Managed Funds n Hedge Funds n Options on Futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Transaction Costs in Forward and Futures Trading
n Commissions n Bid-Ask Spread n Delivery Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets
n Regulation is nearly always at the federal level; e.g., u Commodity Futures Trading Commission (U.S.) u Financial Services Authority (U.K.) u Financial Services Agency (Japan)
n Objective of most federal regulation F ensuring public information available F authorization and licensing of contracts and exchanges F contract approval F market surveillance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets (continued)
n Arbitration of disputes is sometimes done through the federal government and the courts but often through self- regulatory organizations such as the National Futures Association in the U. S.
n Note: Forward markets are regulated only indirectly and, thus, are largely unregulated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
OTC Central Clearing
n Dodd-Frank Act of 2010 further motivated efforts in the OTC derivatives markets for central clearing
n OTC central clearing should provide more transparency to this opaque market and more accountability
n Several clearing corporations are competing for OTC derivatives central clearing
n OTC central clearing is like the spoke and hub system used by some airlines
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Appendix 8: Taxation of Futures Contracts
n Treated as 60 % capital gains and 40 % ordinary income. n Capital gains subject to 28 % maximum. n Must be marked to market at year end. n New single stock futures are taxed the same as individual
stocks. n Hedge transactions covered in Chapter 11.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Price Quotations
n Newspapers (such as The Wall Street Journal)
n Web sites
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Types of Futures Contracts n Agricultural Commodities n Natural Resources n Miscellaneous Commodities n Foreign Currencies n Federal funds and Eurodollars n Treasury Notes and Bonds n Swap Futures n Equities n Managed Funds n Hedge Funds n Options on Futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Transaction Costs in Forward and Futures Trading
n Commissions n Bid-Ask Spread n Delivery Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets
n Regulation is nearly always at the federal level; e.g., u Commodity Futures Trading Commission (U.S.) u Financial Services Authority (U.K.) u Financial Services Agency (Japan)
n Objective of most federal regulation F ensuring public information available F authorization and licensing of contracts and exchanges F contract approval F market surveillance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets (continued)
n Arbitration of disputes is sometimes done through the federal government and the courts but often through self- regulatory organizations such as the National Futures Association in the U. S.
n Note: Forward markets are regulated only indirectly and, thus, are largely unregulated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
OTC Central Clearing
n Dodd-Frank Act of 2010 further motivated efforts in the OTC derivatives markets for central clearing
n OTC central clearing should provide more transparency to this opaque market and more accountability
n Several clearing corporations are competing for OTC derivatives central clearing
n OTC central clearing is like the spoke and hub system used by some airlines
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Appendix 8: Taxation of Futures Contracts
n Treated as 60 % capital gains and 40 % ordinary income. n Capital gains subject to 28 % maximum. n Must be marked to market at year end. n New single stock futures are taxed the same as individual
stocks. n Hedge transactions covered in Chapter 11.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Price Quotations
n Newspapers (such as The Wall Street Journal)
n Web sites
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Types of Futures Contracts n Agricultural Commodities n Natural Resources n Miscellaneous Commodities n Foreign Currencies n Federal funds and Eurodollars n Treasury Notes and Bonds n Swap Futures n Equities n Managed Funds n Hedge Funds n Options on Futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Transaction Costs in Forward and Futures Trading
n Commissions n Bid-Ask Spread n Delivery Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets
n Regulation is nearly always at the federal level; e.g., u Commodity Futures Trading Commission (U.S.) u Financial Services Authority (U.K.) u Financial Services Agency (Japan)
n Objective of most federal regulation F ensuring public information available F authorization and licensing of contracts and exchanges F contract approval F market surveillance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets (continued)
n Arbitration of disputes is sometimes done through the federal government and the courts but often through self- regulatory organizations such as the National Futures Association in the U. S.
n Note: Forward markets are regulated only indirectly and, thus, are largely unregulated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
OTC Central Clearing
n Dodd-Frank Act of 2010 further motivated efforts in the OTC derivatives markets for central clearing
n OTC central clearing should provide more transparency to this opaque market and more accountability
n Several clearing corporations are competing for OTC derivatives central clearing
n OTC central clearing is like the spoke and hub system used by some airlines
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Appendix 8: Taxation of Futures Contracts
n Treated as 60 % capital gains and 40 % ordinary income. n Capital gains subject to 28 % maximum. n Must be marked to market at year end. n New single stock futures are taxed the same as individual
stocks. n Hedge transactions covered in Chapter 11.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Price Quotations
n Newspapers (such as The Wall Street Journal)
n Web sites
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Types of Futures Contracts n Agricultural Commodities n Natural Resources n Miscellaneous Commodities n Foreign Currencies n Federal funds and Eurodollars n Treasury Notes and Bonds n Swap Futures n Equities n Managed Funds n Hedge Funds n Options on Futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Transaction Costs in Forward and Futures Trading
n Commissions n Bid-Ask Spread n Delivery Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets
n Regulation is nearly always at the federal level; e.g., u Commodity Futures Trading Commission (U.S.) u Financial Services Authority (U.K.) u Financial Services Agency (Japan)
n Objective of most federal regulation F ensuring public information available F authorization and licensing of contracts and exchanges F contract approval F market surveillance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets (continued)
n Arbitration of disputes is sometimes done through the federal government and the courts but often through self- regulatory organizations such as the National Futures Association in the U. S.
n Note: Forward markets are regulated only indirectly and, thus, are largely unregulated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
OTC Central Clearing
n Dodd-Frank Act of 2010 further motivated efforts in the OTC derivatives markets for central clearing
n OTC central clearing should provide more transparency to this opaque market and more accountability
n Several clearing corporations are competing for OTC derivatives central clearing
n OTC central clearing is like the spoke and hub system used by some airlines
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Appendix 8: Taxation of Futures Contracts
n Treated as 60 % capital gains and 40 % ordinary income. n Capital gains subject to 28 % maximum. n Must be marked to market at year end. n New single stock futures are taxed the same as individual
stocks. n Hedge transactions covered in Chapter 11.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Price Quotations
n Newspapers (such as The Wall Street Journal)
n Web sites
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Types of Futures Contracts n Agricultural Commodities n Natural Resources n Miscellaneous Commodities n Foreign Currencies n Federal funds and Eurodollars n Treasury Notes and Bonds n Swap Futures n Equities n Managed Funds n Hedge Funds n Options on Futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Transaction Costs in Forward and Futures Trading
n Commissions n Bid-Ask Spread n Delivery Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets
n Regulation is nearly always at the federal level; e.g., u Commodity Futures Trading Commission (U.S.) u Financial Services Authority (U.K.) u Financial Services Agency (Japan)
n Objective of most federal regulation F ensuring public information available F authorization and licensing of contracts and exchanges F contract approval F market surveillance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets (continued)
n Arbitration of disputes is sometimes done through the federal government and the courts but often through self- regulatory organizations such as the National Futures Association in the U. S.
n Note: Forward markets are regulated only indirectly and, thus, are largely unregulated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
OTC Central Clearing
n Dodd-Frank Act of 2010 further motivated efforts in the OTC derivatives markets for central clearing
n OTC central clearing should provide more transparency to this opaque market and more accountability
n Several clearing corporations are competing for OTC derivatives central clearing
n OTC central clearing is like the spoke and hub system used by some airlines
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Appendix 8: Taxation of Futures Contracts
n Treated as 60 % capital gains and 40 % ordinary income. n Capital gains subject to 28 % maximum. n Must be marked to market at year end. n New single stock futures are taxed the same as individual
stocks. n Hedge transactions covered in Chapter 11.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Price Quotations
n Newspapers (such as The Wall Street Journal)
n Web sites
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Types of Futures Contracts n Agricultural Commodities n Natural Resources n Miscellaneous Commodities n Foreign Currencies n Federal funds and Eurodollars n Treasury Notes and Bonds n Swap Futures n Equities n Managed Funds n Hedge Funds n Options on Futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Transaction Costs in Forward and Futures Trading
n Commissions n Bid-Ask Spread n Delivery Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets
n Regulation is nearly always at the federal level; e.g., u Commodity Futures Trading Commission (U.S.) u Financial Services Authority (U.K.) u Financial Services Agency (Japan)
n Objective of most federal regulation F ensuring public information available F authorization and licensing of contracts and exchanges F contract approval F market surveillance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets (continued)
n Arbitration of disputes is sometimes done through the federal government and the courts but often through self- regulatory organizations such as the National Futures Association in the U. S.
n Note: Forward markets are regulated only indirectly and, thus, are largely unregulated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
OTC Central Clearing
n Dodd-Frank Act of 2010 further motivated efforts in the OTC derivatives markets for central clearing
n OTC central clearing should provide more transparency to this opaque market and more accountability
n Several clearing corporations are competing for OTC derivatives central clearing
n OTC central clearing is like the spoke and hub system used by some airlines
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Appendix 8: Taxation of Futures Contracts
n Treated as 60 % capital gains and 40 % ordinary income. n Capital gains subject to 28 % maximum. n Must be marked to market at year end. n New single stock futures are taxed the same as individual
stocks. n Hedge transactions covered in Chapter 11.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Price Quotations
n Newspapers (such as The Wall Street Journal)
n Web sites
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Types of Futures Contracts n Agricultural Commodities n Natural Resources n Miscellaneous Commodities n Foreign Currencies n Federal funds and Eurodollars n Treasury Notes and Bonds n Swap Futures n Equities n Managed Funds n Hedge Funds n Options on Futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Transaction Costs in Forward and Futures Trading
n Commissions n Bid-Ask Spread n Delivery Costs
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets
n Regulation is nearly always at the federal level; e.g., u Commodity Futures Trading Commission (U.S.) u Financial Services Authority (U.K.) u Financial Services Agency (Japan)
n Objective of most federal regulation F ensuring public information available F authorization and licensing of contracts and exchanges F contract approval F market surveillance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets (continued)
n Arbitration of disputes is sometimes done through the federal government and the courts but often through self- regulatory organizations such as the National Futures Association in the U. S.
n Note: Forward markets are regulated only indirectly and, thus, are largely unregulated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
OTC Central Clearing
n Dodd-Frank Act of 2010 further motivated efforts in the OTC derivatives markets for central clearing
n OTC central clearing should provide more transparency to this opaque market and more accountability
n Several clearing corporations are competing for OTC derivatives central clearing
n OTC central clearing is like the spoke and hub system used by some airlines
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Appendix 8: Taxation of Futures Contracts
n Treated as 60 % capital gains and 40 % ordinary income. n Capital gains subject to 28 % maximum. n Must be marked to market at year end. n New single stock futures are taxed the same as individual
stocks. n Hedge transactions covered in Chapter 11.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Futures Price Quotations
n Newspapers (such as The Wall Street Journal)
n Web sites
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Types of Futures Contracts n Agricultural Commodities n Natural Resources n Miscellaneous Commodities n Foreign Currencies n Federal funds and Eurodollars n Treasury Notes and Bonds n Swap Futures n Equities n Managed Funds n Hedge Funds n Options on Futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Transaction Costs in Forward and Futures Trading
n Commissions n Bid-Ask Spread n Delivery Costs
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets
n Regulation is nearly always at the federal level; e.g., u Commodity Futures Trading Commission (U.S.) u Financial Services Authority (U.K.) u Financial Services Agency (Japan)
n Objective of most federal regulation F ensuring public information available F authorization and licensing of contracts and exchanges F contract approval F market surveillance
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Regulation of Futures Markets (continued)
n Arbitration of disputes is sometimes done through the federal government and the courts but often through self- regulatory organizations such as the National Futures Association in the U. S.
n Note: Forward markets are regulated only indirectly and, thus, are largely unregulated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
OTC Central Clearing
n Dodd-Frank Act of 2010 further motivated efforts in the OTC derivatives markets for central clearing
n OTC central clearing should provide more transparency to this opaque market and more accountability
n Several clearing corporations are competing for OTC derivatives central clearing
n OTC central clearing is like the spoke and hub system used by some airlines
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 8: 0
Appendix 8: Taxation of Futures Contracts
n Treated as 60 % capital gains and 40 % ordinary income. n Capital gains subject to 28 % maximum. n Must be marked to market at year end. n New single stock futures are taxed the same as individual
stocks. n Hedge transactions covered in Chapter 11.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Chapter 9: Principles of Pricing Forwards, Futures, and Options on Futures
Futures markets are an accurate representation of consensus opinion, but if we pool all our ignorance, we do not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Important Concepts in Chapter 9
n Price and value of forward and futures contracts n Relationship between forward and futures prices n Determination of the spot price of an asset n Carry arbitrage model for theoretical fair price n Contango, backwardation, and convenience yield n Futures prices and risk premiums n Pricing options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage
n The Concept of Price Versus Value u Normally in an efficient market, price = value. u For a futures or forward, price is the contracted rate of future purchase.
Value is something different. u At the beginning of a contract, value = 0 for both futures and forwards.
n Notation u Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures
contracts created at time 0 and expiring at time T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract u Forward price at expiration: F F(T,T) = ST. F That is, the price of an expiring forward contract is the spot price.
u Value of forward contract at expiration: F VT(0,T) = ST - F(0,T). F An expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (-1) times this.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Forward Contract (continued) u The Value of a Forward Contract Prior to Expiration F A: Go long forward contract at price F(0,T) at time 0. F B: At t go long the asset and take out a loan promising to pay
F(0,T) at T • At time T, A and B are worth the same, ST – F(0,T). Thus,
they must both be worth the same prior to T. • So Vt(0,T) = St – F(0,T)(1+r)-(T-t) • See Table 9.1. F Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 - 100(1.10)-25/365 = 2.65.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n The Value of a Futures Contract u Futures price at expiration: F fT(T) = ST.
u Value during the trading day but before being marked to market: F vt(T) = ft(T) - ft-1(T).
u Value immediately after being marked to market: F vt(T) = 0.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Generic Carry Arbitrage (continued)
n Forward Versus Futures Prices u Forward and futures prices will be equal F One day prior to expiration F More than one day prior to expiration if • Interest rates are certain • Futures prices and interest rates are uncorrelated
u Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
u Default risk can also affect the difference between futures and forward prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities
n Forward and Futures Pricing When the Underlying Generates Cash Flows
u For example, dividends on a stock or index F Assume one dividend DT paid at expiration. F Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT. u For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends. u Dividends reduce the cost of carry. u If D0 represents the present value of the dividends, the model
becomes • f0(T) = (S0 – D0)(1+r)T.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
u For dividends paid at a continuously compounded rate of dc,
u Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
F f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Equities (continued)
n Valuation of Equity Forward Contracts u When there are dividends, to determine the value of a forward contract
during its life F Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t) F where Dt,T is the value at time t of the future dividends to time T
u Or if dividends are continuous,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
u Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
u Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
F Take S0(1+ r)-T units of domestic currency and buy (1+ r)-T units of foreign currency.
F Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
F Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
F Deliver foreign currency and receive F(0,T) units of domestic currency.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
F So an investment of S0(1+ r)-T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ r)-T(1 + r)T F This is called interest rate parity. F Sometimes written as • F(0,T) = S0(1 + r)T/(1 + �)T
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Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u Example (from a European perspective): S0 = €1.0304. U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466.
F F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025 u If forward rate is actually €1.03, then it is overpriced. F Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) =
€1.0161. Sell one forward contract at €1.03. F Earn 5.84% on $0.9861. This grows to $1. F At expiration, deliver $1 and receive €1.03. F Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359) F This transaction is called covered interest arbitrage.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Carry Arbitrage: Currencies (continued)
n Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
u It is also sometimes written as F F(0,T) = S0(1 + �)T(1 + r)-T F Here, the spot rate is being quoted in units of the foreign currency.
u Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
u Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
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(Return to text slide)
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Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
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Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
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Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
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Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
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(Return to text slide)
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Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
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(Return to text slide)
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Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
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Chance/Brooks An Introduction to Derivatives and Risk
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Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
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Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
u First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
F S0 = ST - s - iS0 u If we now allow uncertainty but assume people are risk neutral, we have F S0 = E(ST) - s - iS0
u If we now allow people to be risk averse, they require a risk premium of E(�). Now
F S0 = E(ST) - s - iS0 - E(�)
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Pricing Models and Risk Premiums
n Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
u Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
u Define s + iS0 as the cost of carry. u Denote cost of carry as �. u Note how cost of carry is a meaningful concept only for storable assets
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Pricing Models and Risk Premiums
n The Theoretical Fair Price (Forward/Futures Pricing Revisited)
u Do the following F Buy asset in spot market, paying S0; sell futures contract at price
f0(T); store and incur costs. F At expiration, make delivery. Profit: • P = f0(T) - S0 - q
u This must be zero to avoid arbitrage; thus, • f0(T) = S0 + q
u See Figure 9.2. u Note how arbitrage and quasi-arbitrage make this hold.
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Pricing Models and Risk Premiums
n Forward/Futures Pricing Revisited(continued) u See Figure 9.3 for an illustration of the determination of futures prices. u Contango is f0(T) > S0. See Table 9.2. u When f0(T) < S0, convenience yield is c , an additional return from
holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted. See Table 9.3. Market can be both backwardation and contango. See Table 9.4.
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Pricing Models and Risk Premiums
n Futures Prices and Risk Premia u The no risk-premium hypothesis F Market consists of only speculators. F f0(T) = E(ST). See Figure 9.4.
u The risk-premium hypothesis F E(fT(T)) > f0(T). F When hedgers go short futures, they transfer risk premium to
speculators who go long futures. F E(ST) = f0(T) + E(f). See Figure 9.5.
u Normal contango: E(ST) < f0(T) u Normal backwardation: f0(T) < E(ST)
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Put-Call-Forward/Futures Parity n Can construct synthetic futures with options. n See Table 9.5. n Put-call-forward/futures parity u Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
n Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so T = 35/365 = 0.0959. Risk-free rate at 4.56%.
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Put-Call-Forward/Futures Parity (continued) u So Pe(S0,T,X) = 39 u Ce(S0,T,X) + (X - f0(T))(1+r)-T u = 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70. u Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit
at no risk. Transaction costs would have to be considered.
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Pricing Options on Futures
n The Intrinsic Value of an American Option on Futures u Minimum value of American call on futures F Ca(f0(T),T,X) ³ Max(0, f0(T) - X)
u Minimum value of American put on futures F Pa(f0(T),T,X) ³ Max(0,X - f0(T))
u Difference between option price and intrinsic value is time value.
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Pricing Options on Futures (continued)
n The Lower Bound of a European Option on Futures u For calls, construct two portfolios.
See Table 9.6. u Portfolio A dominates Portfolio B so F Ce(f0(T),T,X) ³ Max[0,(f0(T) - X)(1+r)-T]
u Note that lower bound can be less than intrinsic value even for calls. u For puts, see Table 9.7. u Portfolio A dominates Portfolio B so F Pe(f0(T),T,X) ³ Max[0,(X - f0(T))(1+r)-T]
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Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
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Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
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See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
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Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
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See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
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Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
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See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
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Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
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Chance/Brooks An Introduction to Derivatives and Risk
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See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 9: 0
(Return to text slide)
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
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(Return to text slide)
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See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Pricing Options on Futures (continued)
n Put-Call Parity of Options on Futures u Construct two portfolios, A and B. u See Table 9.8. u The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus, F Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
u Compare to put-call parity for options on spot: F Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T. F If options on spot and options on futures expire at same time,
their values are equal, implying f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
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Pricing Options on Futures (continued)
n Early Exercise of Call and Put Options on Futures u Deep in-the-money call may be exercised early because F behaves almost identically to futures F exercise frees up funds tied up in option but requires no funds to
establish futures F minimum value of European futures call is less than value if it could
be exercised u See Figure 9.6. u Similar arguments hold for puts u Compare to the arguments for early exercise of call and put options on
spot.
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models u Black model for pricing European options on futures
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Chance/Brooks An Introduction to Derivatives and Risk
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Pricing Options on Futures (continued)
n Options on Futures Pricing Models (continued) u Note that with the same expiration for options on spot as options on
futures, this formula gives the same price. u Example F See Table 9.9.
u Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
u For puts
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(Return to text slide)
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See Table 9.10 for a summary of equations. See Figure 9.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
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Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
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Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
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Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
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(Return to text slide)
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(Return to text slide)
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Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
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Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
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Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chapter 10: Futures Arbitrage Strategies Often Jadwin had noted the scene, and unimaginative though he was, had long since conceived the notion of some great, some resistless force within the Board of Trade Building that held the tide of the streets within its grip, alternately drawing it in and throwing it forth. Within there, a great whirlpool, a pit of roaring water spun and thundered, sucking in the life tides of the city, sucking them in as into the mouth of some tremendous cloaca, the maw of some colossal sewer; the vomiting them forth again, spewing them up and out, only to catch them in the return eddy and suck them in afresh.
Frank Norris The Pit, 1902, 1994 edition, Penguin Books, p. 72
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Important Concepts in Chapter 10
n Futures arbitrage strategies n Short-term interest rate futures n Long-term interest rate futures n Stock index futures n Foreign exchange futures
n Cheapest-to-deliver bond n Delivery options
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage
n Cash and Carry/Implied Repo u Cash and carry transaction means to buy asset and sell futures u Repurchase agreement/repo to obtain funding u Overnight vs. term repo u Cost of carry pricing model: f0(t) = S0 + q u Implied repo rate:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Short-Term Interest Rate Arbitrage (continued)
n Cash and Carry/Implied Repo Rate u Also equivalent to buying longer term and converting it to shorter term. u Example. See Table 10.1.
n Eurodollar Arbitrage u Using Eurodollar futures with spot to earn an arbitrage profit. u See Table 10.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Adjustment to futures price using conversion factor, which is the price per $1.00 par of a 6% bond delivered on a particular expiration.
u Invoice price = (Settlement price on position day)(Conversion factor) + Accrued interest
u Example: Delivery on March 2012 contract. Settlement price is 140 ($140,000) on position day.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u You plan to deliver the 5 1/2 of 2028 on March 8. CF = 0.9485. Coupon dates of February 15 and August 15. Last coupon on February 15, 2012. Days from 2/15 to 3/8 is 22. Days from 2/15 to 8/15 is 182. Accrued interest
F $100,000(0.055/2)(22/182) = $332.42 u Invoice price: F $140,000(0.9485) + $332.42 = $133,122.42
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
u On the day of delivery, Thursday, March 8, the short invoices the long $133,122.42. The long pays for and receives the bond on that day.
u Table 10.3 shows CFs and invoice prices for other deliverable bonds on the March 2009 contract.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract
u Recall the option to deliver any T-bond with at least 15 years to maturity or first call.
u Example: Delivery on March 2012 contract of 6 1/4s of May 15, 2030. u Cost of delivering bond F f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Example: On 9/15/11, plan to deliver the 6 1/4s of 5/15/30 on the March 2012 contract on March 8, 2012. f0(T) = 140 , CF = 1.0273, AIt = 2.0890, AIT = 1.9574 (deliver on March 8), B = 148.53125. 114 days between September 15 and March 8. Reinvestment rate = 1.0%.
u Invoice price F 140(1.0273) + 1.9574 = 145.7794
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u There is one interim coupon paid u Forward price of deliverable bond F (148.53125 + 2.0890)(1.01)114/365
- 3.125(1.01)114/365 = 148.2058 u So the bond would cost 2.43 (=148.2058 – 145.7794) more than it would
return.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u All we can do, however, is compare this result with that for another bond. For the 6 1/8ths of November 15, 2027 with CF = 1.0125 and price of 144 5/32, we have accrued interest of 2.0472 on September 15, 2011 and 1.9183 on March 8, 2012. Coupon of 3.0625 on November 15 is reinvested at 1.0% for 114 days and grows to 3.0625(1.01)114/365 = 3.0720.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Forward price is, therefore, F (144.15625 + 2.0472)(1.01)175/365 – 3.0720 = 143.83
u Invoice price is F 142(1.0125) + 1.9183 = 145.69.
u Thus, this bond would produce 1.86 (= 145.69 – 143.83). So the 6 1/8, 11/27 bond is better than the 6 1/4, 5/30 bond.
u Table 10.4 shows these calculations for all deliverable bonds. See Ctd9e.xls for these calculations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Determining the Cheapest-to-Deliver Bond on the Treasury Bond Futures Contract (continued)
u Why identifying the cheapest-to-deliver bond is important: F Identifying the true spot price F Calculating the correct hedge ratio
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
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Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options u The Wild Card Option F Futures market closes at 3:00 pm while spot market stays open until
at least 5:00 pm. F This allows the holder of a short futures contract during the delivery
month to potentially profit from a decline in the price of a deliverable bond during that two hour period in the expiration month.
F Illustration: f3 = futures price at 3:00 pm, B3 = spot price at 3:00 pm. CF = conversion factor
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F Let the short own 1/CF bonds (CF must be > 1.0, so coupon must be
> 6 percent). This is less than one bond per contract so additional bonds, called “the tail,” will have to be purchased in order to make delivery.
F At 5:00 pm, the spot price is B5. It is profitable to purchase these bonds at 5:00 pm if B5 < f3(CF).
F This holds because the invoice price is locked in but the spot price of the bonds can potentially fall.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Wild Card Option (continued) F If the spot price does not fall sufficiently, then the short simply waits
until the next day. By the last eligible delivery day, the short would have to make delivery.
F This is a potentially valuable option granted by the long to the short and its value would have to be reflected in a lower futures price at 3:00 pm.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Quality Option F Also called the switching option, it gives the short the right to change
deliverable bonds if another becomes more attractive. This right also exists in various other futures markets.
F Similar to this is the location option, which is the right to choose from among several eligible delivery locations. This can be valuable when the underlying is a storable commodity.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The End-of-the-Month Option F The right to make delivery any of the business days at the end of the
month after the futures contract has stopped trading, around the third week of the month.
F Similar to the wild card option because the invoice price is locked in when the futures stops trading.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Delivery Options (continued) u The Timing Option F The right to deliver on any eligible day of the delivery month. F Delivery will be made early in the month if the bond earns a coupon
that is less than the cost of financing. F Delivery will be made late in the month if the bond earns a coupon
that exceeds the cost of financing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry u Buy spot T-bond, sell futures. u This will produce a return (implied repo rate) of
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Example: On September 15, 2011, CTD bond on March contract is 6 3/8s
maturing on August 15, 2027. Spot price is 147 1/4, accrued interest is 0.5370, CF = 1.0370 and futures price is 142. From September 15 to March 8 is 175 days so T = 175/365 = 0.47945. There one coupon payment made with a future value of 3.1894 ((6.375/2)(1.01)(22/365). Accrued interest on March 8th, AIT = 0.3853
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Implied Repo/Cost of Carry (continued) u Implied repo rate is, therefore,
u If the bond can be financed in the repo market for less than this rate, then the arbitrage would be profitable. Obviously that is not the case here.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate u Let time t be expiration of nearby futures and T be expiration of deferred
futures. u Go long the nearby and short the deferred. u When nearby expires, take delivery and hold until expiration of deferred.
This creates a forward transaction beginning at t and ending at T
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Example: On September 15, 2011 CTD was 6 3/8s maturing in 8/2026. Examine the March-June spread. March priced at f0(t) = 142 with CF(t) = 1.0370. June priced at f0(T) = 141 with CF(T) = 1.0368. AIt (March 8) = 0.3853 and AIT (June 8) = 1.9966. No coupons in the interim so CIt,T = 0. From March 8 to June 8 is 92 days.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Intermediate and Long-Term Interest Rate Arbitrage (continued)
n Treasury Bond Spread/Implied Repo Rate (continued) u Implied repo rate
u Compare to actual repo rate and note that this is a forward rate. u Note the turtle trade: Implied repo rate on T-bond spread to Fed funds
futures rate
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage n Stock Index Arbitrage u Recall the stock index futures pricing model
u Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend yield is 3% and time to expiration is 40 days so T = 40/365 = 0.1096. Futures should be at
F 1305e(0.052 - 0.03)(0.1096) = 1308.15 u Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the stocks and buy back the futures.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Now find the implied repo rate. Let f0(T) be the actual futures price.
Then
u In our example, this is
u So if you could get financing at less than this rate, the arbitrage would be worth doing.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Stock Index Arbitrage (continued)
n Stock Index Arbitrage (continued) u Some practical considerations F buying and selling all stocks simultaneously F buying fractional contracts F transaction costs of about 0.005 % of spot value.
u Program trading. u See Table 10.5 for stock index arbitrage example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage
n Foreign exchange futures pricing model (annual compounding)
f0(T) = S0(l + r)T/(l + �)T
n Foreign exchange futures pricing model (continuous compounding)
f0(T) = S0e(rc - �c)T Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 10: 0
Foreign Exchange Arbitrage (continued)
n Example: Let spot rate for dollars = 0.7908 euros, U. S. risk-free rate is 5.84%, euro risk-free rate is 3.59% and time to expiration is 90 days so T = 90/365 = 0.2466. Futures should be priced at
u 0.7908e(0.0584 - 0.0359)(0.2466) = 0.7866 euros n Now let the actual futures price be 0.80 euros. This is too high so sell
the futures and buy the index. Hold until expiration. Sell the euros and buy back the futures for an arbitrage profit.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor
n Determine maturity in years (YRS), months (MOS) and days as of first date of expiration month. Use first call date if callable. Ignore days. Let c be coupon rate. Round months down to 0, 3, 6, or 9. Call this MOS*.
u If MOS* = 0,
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Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
u If MOS* = 3,
u If MOS* = 6,
u If MOS* = 9,
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Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
n Example: 5 1/4s of February 15, 2029 delivered on March 2012 contract. On March 1, 2012 remaining life is 16 years, 11 months, 14 days. YRS = 16, MOS = 11. Round down so that MOS* = 9. Find CF6:
n Then CF9 is CF9 = (0.922128 + 0.0525/2)(1.03)-0.5 – 0.0525/4 = 0.9213
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage (continued)
n Example: Let spot rate for dollars = 0.7908 euros, U. S. risk-free rate is 5.84%, euro risk-free rate is 3.59% and time to expiration is 90 days so T = 90/365 = 0.2466. Futures should be priced at
u 0.7908e(0.0584 - 0.0359)(0.2466) = 0.7866 euros n Now let the actual futures price be 0.80 euros. This is too high so sell
the futures and buy the index. Hold until expiration. Sell the euros and buy back the futures for an arbitrage profit.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor
n Determine maturity in years (YRS), months (MOS) and days as of first date of expiration month. Use first call date if callable. Ignore days. Let c be coupon rate. Round months down to 0, 3, 6, or 9. Call this MOS*.
u If MOS* = 0,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
u If MOS* = 3,
u If MOS* = 6,
u If MOS* = 9,
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Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
n Example: 5 1/4s of February 15, 2029 delivered on March 2012 contract. On March 1, 2012 remaining life is 16 years, 11 months, 14 days. YRS = 16, MOS = 11. Round down so that MOS* = 9. Find CF6:
n Then CF9 is CF9 = (0.922128 + 0.0525/2)(1.03)-0.5 – 0.0525/4 = 0.9213
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage (continued)
n Example: Let spot rate for dollars = 0.7908 euros, U. S. risk-free rate is 5.84%, euro risk-free rate is 3.59% and time to expiration is 90 days so T = 90/365 = 0.2466. Futures should be priced at
u 0.7908e(0.0584 - 0.0359)(0.2466) = 0.7866 euros n Now let the actual futures price be 0.80 euros. This is too high so sell
the futures and buy the index. Hold until expiration. Sell the euros and buy back the futures for an arbitrage profit.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor
n Determine maturity in years (YRS), months (MOS) and days as of first date of expiration month. Use first call date if callable. Ignore days. Let c be coupon rate. Round months down to 0, 3, 6, or 9. Call this MOS*.
u If MOS* = 0,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
u If MOS* = 3,
u If MOS* = 6,
u If MOS* = 9,
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Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
n Example: 5 1/4s of February 15, 2029 delivered on March 2012 contract. On March 1, 2012 remaining life is 16 years, 11 months, 14 days. YRS = 16, MOS = 11. Round down so that MOS* = 9. Find CF6:
n Then CF9 is CF9 = (0.922128 + 0.0525/2)(1.03)-0.5 – 0.0525/4 = 0.9213
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage (continued)
n Example: Let spot rate for dollars = 0.7908 euros, U. S. risk-free rate is 5.84%, euro risk-free rate is 3.59% and time to expiration is 90 days so T = 90/365 = 0.2466. Futures should be priced at
u 0.7908e(0.0584 - 0.0359)(0.2466) = 0.7866 euros n Now let the actual futures price be 0.80 euros. This is too high so sell
the futures and buy the index. Hold until expiration. Sell the euros and buy back the futures for an arbitrage profit.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor
n Determine maturity in years (YRS), months (MOS) and days as of first date of expiration month. Use first call date if callable. Ignore days. Let c be coupon rate. Round months down to 0, 3, 6, or 9. Call this MOS*.
u If MOS* = 0,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
u If MOS* = 3,
u If MOS* = 6,
u If MOS* = 9,
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
n Example: 5 1/4s of February 15, 2029 delivered on March 2012 contract. On March 1, 2012 remaining life is 16 years, 11 months, 14 days. YRS = 16, MOS = 11. Round down so that MOS* = 9. Find CF6:
n Then CF9 is CF9 = (0.922128 + 0.0525/2)(1.03)-0.5 – 0.0525/4 = 0.9213
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Foreign Exchange Arbitrage (continued)
n Example: Let spot rate for dollars = 0.7908 euros, U. S. risk-free rate is 5.84%, euro risk-free rate is 3.59% and time to expiration is 90 days so T = 90/365 = 0.2466. Futures should be priced at
u 0.7908e(0.0584 - 0.0359)(0.2466) = 0.7866 euros n Now let the actual futures price be 0.80 euros. This is too high so sell
the futures and buy the index. Hold until expiration. Sell the euros and buy back the futures for an arbitrage profit.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 10: 0
Appendix 10: Determining the CBOT Treasury Bond Conversion Factor
n Determine maturity in years (YRS), months (MOS) and days as of first date of expiration month. Use first call date if callable. Ignore days. Let c be coupon rate. Round months down to 0, 3, 6, or 9. Call this MOS*.
u If MOS* = 0,
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Chance/Brooks An Introduction to Derivatives and Risk
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Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
u If MOS* = 3,
u If MOS* = 6,
u If MOS* = 9,
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Appendix 10: Determining the CBOT Treasury Bond Conversion Factor (continued)
n Example: 5 1/4s of February 15, 2029 delivered on March 2012 contract. On March 1, 2012 remaining life is 16 years, 11 months, 14 days. YRS = 16, MOS = 11. Round down so that MOS* = 9. Find CF6:
n Then CF9 is CF9 = (0.922128 + 0.0525/2)(1.03)-0.5 – 0.0525/4 = 0.9213
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Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
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Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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(Return to text slide)
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
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S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
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Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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(Return to text slide)
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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(Return to text slide)
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
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(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Chapter 11: Forward and Futures Hedging, Spread, and Target Strategies
The beauty of finance and speculation was that they could be different things to different men. To some: poetry or high drama; to others, physics, scientific and immutable; to still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Important Concepts in Chapter 11
n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures hedges
n Examples of spread and target strategies
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? n The value of the firm may not be independent of financial
decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures
contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs.
n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge their market-making activities in
derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market
conditions u There is no such thing as a hedge. Any hedge is an act of taking a position
that an adverse market movement will occur. This, itself, is a form of speculation.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because the hedger owns an asset and plans to sell
it later. u Long hedges can occur because the hedger plans to purchase an asset later. u An anticipatory hedge is a hedge of a transaction that is expected to occur
in the future. u See Table 11.1 for hedging situations.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis F P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from
futures market) F P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from
futures market) F If hedge is closed prior to expiration,
P (short hedge) = St - S0 - (ft - f0) F If hedge is held to expiration, St = ST = fT = ft.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
S0 spot price today f 0 futures price today St spot price at time t prior to expiration f t futures price at time t prior to expiration ST spot price at expiration f T futures price at expiration Π profit from a given strategy
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.
u Basis definition F initial basis: b0 = S0 - f0 F basis at time t: bt = St - ft F basis at expiration: bT = ST - fT = 0
u For a position closed at t: F P (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued) n The Basis (continued) u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will
be f0 + (St - ft). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u All of this reverses for a long hedge. u See Table 11.2 for hedging profitability and the basis.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n The Basis (continued) u Example: March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note: b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit should be change in basis or 1.45.
F At expiration, let ST = $1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35. Buy back futures at $1,408.50, a profit of −19.90. Net gain =1.45 or $145 on 100 oz. of gold.
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. F Profit on spot = −9.63. Profit on futures = 9.97. F Net gain = 0.34 or $34 on 100 oz. F Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34. u Behavior of the basis, see Figure 11.1. u In forward markets, the hedge is customized so there is no basis risk.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Concepts (continued)
n Contract Choice u Which futures underlying asset? F High correlation with spot F Favorably priced
u Which expiration? F The futures with maturity closest to but after the hedge termination
date subject to the suggestion not to be in a contract in its expiration month
F See Table 11.3 for example of recommended contracts for T-bond hedge
F Concept of rolling the hedge forward
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question.
See Table 11.4. • worst case scenario method • current spot position method • anticipated future spot transaction method
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Concepts (continued)
n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio
n Hedge ratio: The number of futures contracts to hedge a particular exposure
u Naïve hedge ratio u Appropriate hedge ratio should be F Nf = −DS/Df F Note that this ratio must be estimated.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio u Profit from short hedge: F P = DS + DfNf
u Variance of profit from short hedge: F sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf
u The optimal (variance minimizing) hedge ratio is F Nf = −sDSDf/sDf2 F This is the beta from a regression of spot price change on futures
price change.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Minimum Variance Hedge Ratio (continued) u Hedging effectiveness is F e* = (risk of unhedged position − risk of hedged position)/risk of
unhedged position F This is coefficient of determination from regression.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at
B:
F where CPt is the cash payment at time t and yB is the yield, or discount rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio (continuation) u An approximation to the change in price for a yield change is
u with DURB being the bond’s duration, which is a weighted- average of the times to each cash payment date on the bond, and � represents the change in the bond price or yield.
u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.
u Modified duration (MD) measures the bond percentage price change for a given change in yield.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continuation) u The hedge ratio is as follows:
u Where MDB » −(DB/B) /DyB and MDf » −(Df/f) /Dyf
u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Price Sensitivity Hedge Ratio (continued) u Alternatively, F Nf = −(Yield beta)PVBPB/PVBPf • where Yield beta is the beta from a regression of spot bond yield
on futures yield and • PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Determination of the Hedge Ratio (continued)
n Stock Index Futures Hedging u Appropriate hedge ratio is F Nf = −(bS/bf)(S/f) F where bS is the beta from the CAPM and bf is the beta of the futures,
often assumed to be 1.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies
n Long Hedge With Foreign Currency Futures u American firm planning to buy foreign inventory and will pay in foreign
currency. u See Table 11.5.
n Short Hedge With Foreign Currency Forwards u British subsidiary of American firm will convert pounds to dollars. u See Table 11.6.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Hedges u First let us look at the CBOT T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first
call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6%
coupon. Adjustments, explained in Chapter 10, are made to reflect other coupons.
F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is $93.4375.
F Contract size is $100,000 face value so price is $93,437.50
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Intermediate and Long-Term Interest Rate Hedges (continued)
u Hedging a Long Position in a Government Bond F See Table 11.7 for example.
u Anticipatory Hedge of a Future Purchase of a Treasury Note F See Table 11.8 for example.
u Hedging a Corporate Bond Issue F See Table 11.9 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Management, 9th ed. Ch. 11: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
(Return to text slide)
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Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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(Return to text slide)
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Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
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Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
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(Return to text slide)
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Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
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(Return to text slide)
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Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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(Return to text slide)
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Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
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(Return to text slide)
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Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
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(Return to text slide)
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(Return to text slide)
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Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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(Return to text slide)
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Hedging Strategies (continued)
n Stock Market Hedges u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined
by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000
u Stock Portfolio Hedge F See Table 11.10 for example.
u Anticipatory Hedge of a Takeover F See Table 11.11 for example.
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(Return to text slide)
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(Return to text slide)
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Spread Strategies n Intramarket Spreads u Based on changes in the difference in carry costs u See Figure 11.2 for illustration.
n Treasury Bond Futures Spreads u See Figure 11.3 and Figure 11.4 for illustration the relationship between
changes in spreads and interest rates. u See Table 11.12 for calculation of Tbond futures spread profits.
n See Figure 11.5 for illustration of stock index spreads
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Intermarket Spread Strategies
n Intermarket spread strategies involve two futures contracts on different underlying instruments
n Intermarket spread strategies tend to be more risky than intramarket spreads because there is both the change in spreads and the change in underlying instruments
n NOB denotes notes over bonds n Intermarket spread strategies could also involve various
equity markets
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Target Strategies: Bonds
n Target Duration with Bond Futures u Number of futures needed to change modified duration
u Goal is to move the modified duration from its current value to a new target value
u See Table 11.13 for illustration.
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(Return to text slide)
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Target Strategies: Equities
n Alpha Capture u Number of futures to hedge systematic risk
u Goal is to move the eliminate systematic risk u See Table 11.14 for illustration.
n Target Beta (see Table 11.15 for illustration.)
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(Return to text slide)
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(Return to text slide)
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Target Strategies: Equities (continued)
n Tactical Asset Allocation u Strategic asset allocation – long run target weights for each asset class u Tactical asset allocation – short run deviations in weights for each asset
class u See Table 11.16 for illustration.
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(To Continue) (Return to text slide 31)
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(Return to text slide 31)
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Summary n Table 11.17 recaps the types of hedge situations, the nature
of the risk and how to hedge the risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Appendix 11: Taxation of Hedging
n Hedges used by businesses to protect inventory and in standard business transactions are taxed as ordinary income.
n Transactions must be shown to be legitimate hedges and not just speculation outside of the norm of ordinary business activities. This is called the business motive test.
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(Return to text slide 31)
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Summary n Table 11.17 recaps the types of hedge situations, the nature
of the risk and how to hedge the risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Appendix 11: Taxation of Hedging
n Hedges used by businesses to protect inventory and in standard business transactions are taxed as ordinary income.
n Transactions must be shown to be legitimate hedges and not just speculation outside of the norm of ordinary business activities. This is called the business motive test.
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(To Continue) (Return to text slide 31)
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(To Previous Slide)
(Return to text slide 31)
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Summary n Table 11.17 recaps the types of hedge situations, the nature
of the risk and how to hedge the risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Appendix 11: Taxation of Hedging
n Hedges used by businesses to protect inventory and in standard business transactions are taxed as ordinary income.
n Transactions must be shown to be legitimate hedges and not just speculation outside of the norm of ordinary business activities. This is called the business motive test.
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(To Continue) (Return to text slide 31)
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(Return to text slide 31)
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Summary n Table 11.17 recaps the types of hedge situations, the nature
of the risk and how to hedge the risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Appendix 11: Taxation of Hedging
n Hedges used by businesses to protect inventory and in standard business transactions are taxed as ordinary income.
n Transactions must be shown to be legitimate hedges and not just speculation outside of the norm of ordinary business activities. This is called the business motive test.
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Summary n Table 11.17 recaps the types of hedge situations, the nature
of the risk and how to hedge the risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 11: 0
Appendix 11: Taxation of Hedging
n Hedges used by businesses to protect inventory and in standard business transactions are taxed as ordinary income.
n Transactions must be shown to be legitimate hedges and not just speculation outside of the norm of ordinary business activities. This is called the business motive test.
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Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
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Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
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n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
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Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
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Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
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Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
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Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Chapter 12: Swaps
Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants.
Aaron Brown Risk Professional, April 2010, p. 18
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Important Concepts in Chapter 12
n The concept of a swap n Different types of swaps, based on underlying currency,
interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
n Definition of a swap n Four types of swaps u Currency u Interest rate u Equity u Commodity (not covered in this book)
n Characteristics of swaps u No cash up front u Notional amount u Settlement date, settlement period u Credit risk u Dealer market
n See Figure 12.1 for growth in world-wide notional amount n See Figure 12.2 for growth in world-wide gross market value
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps
u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million notional
amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are
F Payments are netted. F See Figure 12.3 for payment pattern F See Table 12.1 for sample of payments after-the-fact.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps F How is the fixed rate determined? F A digression on floating-rate securities. The price of a LIBOR zero
coupon bond for maturity of ti days is
• Starting at the maturity date and working back, we see that the price is par on each coupon date. See Figure 12.4.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F By adding the notional amounts at the end, we can separate the cash
flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond.
F See Figure 12.5. F The value of a fixed-rate bond (q = days/360):
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F The value of a floating-rate bond
F At time t, between 0 and 1,
F The value of the swap (pay fixed, receive floating) is, therefore,
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Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F To price the swap at the start, set this value to zero and solve
for R
F See Table 12.2 for an example. F Note how dealers quote as a spread over Treasury rate. F To value a swap during its life, simply find the difference
between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk.
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Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
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Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
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Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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(Return to text slide)
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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(Return to text slide)
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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(Return to text slide)
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
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Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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(Return to text slide)
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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(Return to text slide)
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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(Return to text slide)
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(Return to text slide)
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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(Return to text slide)
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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(Return to text slide)
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(Return to text slide)
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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(Return to text slide)
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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(Return to text slide)
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(Return to text slide)
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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(Return to text slide)
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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(Return to text slide)
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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(Return to text slide)
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(Return to text slide)
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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(Return to text slide)
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
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(Return to text slide)
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Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Interest Rate Swaps (continued)
u The Pricing and Valuation of Interest Rate Swaps (continued) F A basis swap is equivalent to the difference between two plain vanilla
swaps based on different rates: • A swap to pay T-bill, receive fixed, plus • A swap to pay fixed, receive LIBOR, equals • A swap to pay T-bill, receive LIBOR, plus pay the difference
between the LIBOR and T-bill fixed rates • See Tables 12.4 and Table 12.5 for examples of pricing and
valuation of a basis swap.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaps (continued)
u Interest Rate Swap Strategies F See Figure 12.6 for example of converting floating-rate loan into
fixed-rate loan F Other types of swaps • Index amortizing swaps • Diff swaps • Constant maturity swaps
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(Return to text slide)
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Currency Swaps
u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged.
F See Figure 12.7. u Note the relationship between interest rate and currency swaps in Figure
12.8.
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps F Let dollar notional amount be NP$. Then euro notional
amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
F For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€.
F No pricing is required for the floating side of a currency swap. F See Table 12.6. F During the life of the swap, we value it by finding the
difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
F See Table 12.7 for calculations of values of streams of payments per unit notional amount.
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(Return to text slide)
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Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014 F Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579 F Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308 F Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Currency Swaps (continued) u Pricing and Valuation of Currency Swaps (continued) F Value of swap to pay € fixed, receive $ fixed • $9,915,014 - €10,088,308($0.9790/€) = $38,560 F Value of swap to pay € fixed, receive $ floating • $9,932,579 - €10,088,308($0.9790/€) = $56,125 F Value of swap to pay € floating, receive $ fixed • $9,915,014 - €10,091,157($0.9790/€) = $35,771 F Value of swap to pay € floating, receive $ floating • $9,932,579 - €10,091,157($0.9790/€) = $53,336
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Currency Swaps (continued) u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to
borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk.
F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional amounts.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps u Characteristics F One party pays the return on an equity, the other pays fixed, floating,
or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed
F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 F See Table 12.8 for example of payments. The first equity payment is
F So the first net payment is IVM pays $285,657.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Equity Swaps (continued) u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be
F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps F For a swap to pay fixed and receive equity, we replicate as
follows: • Invest $1 in stock • Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360. – On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1 – Owe interest of Rq – Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over. Reinvest in the stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F On day 360, stock is worth S360/S180. F Liquidate stock. Pay back loan of $1 and interest of Rq. F Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the
second swap payment. F The value of the position is the value of the swap. In general for n
payments, the value at the start is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Setting the value to zero and solving for R gives
F which is the same as the fixed rate on an interest rate swap. See Table 12.9 for pricing the IVM swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F To value the swap at time t during its life, consider the party
paying fixed and receiving equity. F To replicate the first payment, at time t • Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date. • At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap).
• Do this for each payment on the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 12: 0
Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F The cost to do this strategy at time t is
F This is the value of the swap. See Table 12.10 for an example of the IVM swap.
F To value the equity swap receiving floating and paying equity, note the equivalence to
• A swap to pay equity and receive fixed, plus • A swap to pay fixed and receive floating. F So we can use what we already know.
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is • 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884 F The value of the floating payments (plus hypothetical notional
amount) is • (1 + 0.03(90/360))(0.9971) = 1.00457825 F The plain vanilla swap value is, thus, • 1.00457825 – 1.00159884 = 0.00297941 F For a $25 million notional amount, • $25,000,000(0.00297941) = $74,485 F So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed) • -$227,964 + $74,485 = -$153,479
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For swaps to pay one equity and receive another, replicate by selling
short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.
F For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap is
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Equity Swaps (continued) u Pricing and Valuation of Equity Swaps (continued) F For $25 million notional amount, the value is • $25,000,000(0.03312974) = $828,244
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Equity Swaps (continued) u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.10 for example. F Some risks • default • tracking error • cash flow shortages
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Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap
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Summary
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Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
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Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
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n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
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(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
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n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
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(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
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(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
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Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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(Return to text slide)
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Chapter 13: Interest Rate Forwards and Options
As with a second-hand car, you never really know what an OTC option is worth until you actually sell it or buy it. Placing a value on it in the interim is, in some ways, only a more sophisticated version of pinning the tail on the donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Important Concepts in Chapter 13
n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward swaps
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond
relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate
relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case
of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a derivative on a bond
n See Figure 13.1 for notional amount of FRAs and interest rate options over time.
n See Figure 13.2 for gross market value of FRAs and interest rate options over time.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate
u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted.
For FRA on m-day LIBOR, the payoff is
F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional amount of $20 million. Agreed upon rate is 5 percent. Payoff will be
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued)
F Some possible payoffs. If LIBOR at expiration is 4 percent,
F So the long has to pay $49,505. If LIBOR at expiration is 6 percent, the payoff is
F Note the terminology of FRAs: A � B means FRA expires in A months and underlying matures in B months.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the
underlying be an m-day rate. L0(h) is spot rate on day 0 for h days, L0(h+m) is spot rate on day 0 for h + m days. Assume notional amount of $1.
F To find the fixed rate, we must replicate an FRA: • Short a Eurodollar maturing in h+m days that pays 1 + F(m/360).
This is a loan that can be paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays $1
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, • Loan we owe has a market value of
• Pay if off early. Collect $1 on the ED we hold. So total cash flow is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) • This can be rearranged to get
F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F:
F This is just the forward rate in the LIBOR term structure. See Table 13.1 for an example.
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Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g.
If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is
F See Table 13.2 for example.
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Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date
on which they are exposed to interest rate risk. F See Table 13.3 and Figure 13.3 for an example. F Note that a series of FRAs is similar to a swap; however, in a swap all
payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.
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Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Management, 9th ed. Ch. 13: 0
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Management, 9th ed. Ch. 13: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Management, 9th ed. Ch. 13: 0
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Management, 9th ed. Ch. 13: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Management, 9th ed. Ch. 13: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
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Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Management, 9th ed. Ch. 13: 0
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Management, 9th ed. Ch. 13: 0
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Management, 9th ed. Ch. 13: 0
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Management, 9th ed. Ch. 13: 0
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Interest Rate Options u Definition: an option in which the underlying is an interest rate; it
provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment.
u The fixed rate is called the exercise rate. u Most are European-style.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is
F The payoff of an interest rate put is
F The payoff occurs m days after expiration. F Example: notional amount of $20 million, expiration in 30 days,
underlying of 90-day LIBOR, exercise rate of 5 percent.
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Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 1 percent at expiration, payoff of a call is
F The payoff of a put is
F If LIBOR is 9 percent at expiration, payoff of a call is
F The payoff of a put is
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Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is
sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at
the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional amount)
(days/360). F See Table 13.4 for illustration.
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Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5 and Figure 13.4 for an example of the use of an interest
rate call by a borrower to hedge an anticipated loan. F See Table 13.6 and Figure 13.5 for an example of the use of an interest
rate put by a lender to hedge an anticipated loan.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a
floating-rate loan is called an interest rate cap. The component calls are referred to as caplets.
F A combination of interest rate puts used by a lender to hedge a floating- rate loan is called an interest rate floor. The component puts are referred to as floorlets.
F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap • Each component caplet pays off independently of the others. • See Table 13.7 for an example of a borrower using an interest rate
cap. • To price caps, price each component caplet individually and add up
the prices of the caplets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor • Each component floorlet pays off independently of the others • See Table 13.8 for an example of a lender using an interest rate floor. • To price floors, price each component floorlet individually and add
up the prices of the floorlets.
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Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars • A borrower using a long cap can combine it with a short floor so that
the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.
• See Table 13.9 for example.
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Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
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Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
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Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
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Summary
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Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
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Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 13: 0
(Return to text slide)
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Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
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Summary
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Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
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Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
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Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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(Return to text slide)
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Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
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Chance/Brooks An Introduction to Derivatives and Risk
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Summary
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Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the
payoff of a long call is • 0 if LIBOR � X • LIBOR – X if LIBOR > X F The payoff of a short put is • – (X – LIBOR) if LIBOR � X • 0 if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed
rate, the long cap and short floor replicate the swap.
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Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed-rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year
swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929) – 720 day rate: 0.1328 (0.7901) – 1080 day rate: 0.1451 (0.6967)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) • Exercise would create a stream of 11.5 percent fixed payments and
LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional amount. The value of this stream of payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is
F The value of a receiver swaption at expiration is
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Interest Rate Swaptions and Forward Swaps (continued)
u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the
market, R, into the formula for the payoff of a swaption gives • Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967)) F This is the formula for the payoff of a put option on a bond with 11.5 percent
coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.
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Swaption and Callable Bonds u One application of swaptions relates to callable bonds u Recall callable bond issuer has sold (issued) bonds and purchased a call
option u A receiver swaption is comparable to the embedded call option of a bond F See Figure 13.6
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the
previous result, we would price swaptions using models for pricing options on bonds.
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits
the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a
three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) • 360 days: 0.09 (0.9174) • 720 days: 0.1006 (0.8325) • 1080 days: 0.1103 (0.7514) • 1440 days: 0.12 (0.6757) • 1800 days: 0.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three
years.
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued)
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices
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Interest Rate Swaptions and Forward Swaps (continued)
u Forward Swaps (continued) F The rate on the forward swap would be
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Chance/Brooks An Introduction to Derivatives and Risk
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Interest Rate Swaptions and Forward Swaps (continued)
u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used • To exit a swap • As a substitute for an option on a bond • Creating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no
cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.
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Summary
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Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
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Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
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Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
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Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
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Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
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Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
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(Return to text slide)
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Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
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(Return to text slide)
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Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
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Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
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(Return to text slide)
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Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
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(Return to text slide)
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Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Chapter 14: Advanced Derivatives and Strategies
Blinded by greed and wishful thinking we often seem to believe that huge and growing market momentum is a strong signal that a pattern will continue. In fact, such momentum often creates the very conditions that produce a painful correction - so-called self-referential risk. The growth and eventual unraveling of the sub-prime mortgage market is just the latest example.
David Rowe
Risk, December 2008, p. 93
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Important Concepts in Chapter 14 n The concept of portfolio insurance and its execution using
puts, calls, futures and risk-free debt n New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities
n Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options
n Derivatives on electricity and weather
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies
n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For
a portfolio worth V, we should hold F N = V/(S0 + P) puts and shares
u This will establish a minimum of F Vmin = XV/(S0 + P) where X is the exercise price
u Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99% continuously compounded. Volatility is 0.155.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for
V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57)
= 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 F This guarantees a minimum return of
1.0014(365/84) - 1 = .0061 per year, which must be below the risk- free rate.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – 92,036 shares worth 510 = $46,938,360 – 92,036 puts worth $0 = $0 – Total value = $46,938,360 (> Vmin) • Index is 450 at expiration – Sell stock by exercising puts so you have 92,036(485) =
$44,637,460 (» Vmin)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F See Figure 14.1.
u If calls and risk-free debt used, F NB = Vmin/BT (number of units of debt) F NC = V/(S0 + P) (number of calls) F So NB = 44,637,585/100 = 446,376 F NC = 92,036
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) F Outcomes • Index is 510 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $25 = $2,300,900 – Total value = $46,938,500 (> Vmin) • Index is 450 at expiration – Risk-free debt worth $44,637,600 – 92,036 calls worth $0 – Total value = $44,637,600 (» Vmin) • See Figure 14.2.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and
futures or stock and risk-free debt that can replicate the stock-put or call- debt.
F This can be easier because the futures and debt markets are more liquid than the options markets
F The number of futures required is
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Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Portfolio Insurance (continued) u Alternatively, use stock and risk-free debt:
u See Table 14.1 for example of dynamic hedge
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards u Forward contracts on stock or stock indices u Precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At
expiration, however, its value can be negative, unlike an ordinary call. F See Table 14.2. Note that K = compound future value of call with
exercise price F plus compound future value of stock, which is forward price of stock.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) F Example using DCRB: S0 = 125.94, T = 0.0959, rc = 0.0446,
volatility = 0.83. • F = 125.94e0.0446(0.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e0.0446(0.0959) = 139.41 • See Figure 14.3. F Note similarity to forward contract and call option. F To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Advanced Equity Derivatives and Strategies (continued)
n Equity Forwards (continued) u For example, 15 days later, DCRB is at 115.75,
T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain
u A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
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Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
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Chance/Brooks An Introduction to Derivatives and Risk
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Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
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(Return to text slide)
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Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock
Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a
foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any
increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent
of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 with a volatility of 0.12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Equity Derivatives and Strategies (continued)
n Equity-Linked Debt (continued) ● Option payoff is $10(0.5)Max(0,(ST - 1500)/1500). This can be written
as F (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with
exercise price 1500. ● Plugging values into Black-Scholes-Merton model gives call value of
$96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worthwhile to some investors.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives n Structured Notes u Definition: an intermediate term debt security issued by a corporation
with a good credit rating in which the coupon is altered by the use of a derivative. Examples:
F Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate).
F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise.
F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Structured Notes (continued) u Example: An issuer could hedge it by a swap paying LIBOR and
receiving fixed rate F LIBOR < 12: – (12 – LIBOR) (note) + Fixed rate
– LIBOR (swap) = Fixed rate – 12 F LIBOR � 12: 0 (note) + Fixed rate – LIBOR (swap) = Fixed rate –
LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.
u Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a
process called securitization. u Mortgage-backed securities are subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the
principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a
mortgage pass-through. Called principal only (PO) or interest only (IO).
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single
$100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity.
F See Table 14.3 for amortization schedule. Annual payment would be • $100,000/[(1– (1.0975)-30)/0.0975] = $10,387.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in
year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … +
8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2
+ … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690
= $120,944
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F Let discount rate drop to 6% and assume homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 = $17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 + $98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued)
F If the discount rate rises to 8% and there is no change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($1,773 + $86,574)(1.08)-12 = $42,128, a 10% loss
• Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) ● If rate goes to 8% and prepayment moves back to year 14, F Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2
+ . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% F Value of PO = $637(1.08)-1 + $699(1.08)-2
+ . . . + ($2,136 + $82,492)(1.08)-14 = $37,276, a loss of 20%.
F Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are typically very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces
called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some
receive any residual after other tranches have been repaid.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Advanced Interest Rate Derivatives (continued)
n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments
according to different priorities. F Some CMO tranches are extremely volatile and others have low
volatility. F A CMO is generally a fairly complex security.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options
n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types: F Asset-or-nothing options pay the holder the asset if the option expires
in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash
(usually $1) if the option expires in the money and nothing otherwise.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) ● See Table 14.4 for example of a portfolio short cash-or-nothings and long
X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
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Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Management, 9th ed. Ch. 14: 0
Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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Chance/Brooks An Introduction to Derivatives and Risk
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(Return to text slide)
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Exotic Options (continued)
n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return
Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is 0.11 and time to expiration is 0.5 years. We obtain
F d1 = 0.3526, N(0.35) = 0.6368 F DCaon = 1440(0.6368) = 917
u For 1,440 cash-or-nothing options, F d2 = 0.2748, N(0.27) = 0.6064 F (1,440)DCcon = 1440e-0.0488(0.5)(0.6064) = 852.17.
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Exotic Options (continued)
n Digital and Chooser Options (continued) u A variation of the previously covered pay-later option is the contingent-
pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so
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(Return to text slide)
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Exotic Options (continued) n Digital and Chooser Options (continued) u Solving for Ccp gives
u For the example we have been using
u Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is
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Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also called as-you-like-it options, they enable the
investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put.
F Assume the decision must be made at time t < T F The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) F Compare and contrast chooser with straddle.
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Exotic Options (continued) n Digital and Chooser Options (continued) u Example: DCRB chooser in which choice must be made in 20 days.
Call/put expires in 35 days. S0 = 125.94, X = 125, � = 0.83, rc = 0.0446. T = 35/365 = 0.0959, t = 20/365 = 0.0548 so T - t = 0.0959 - 0.0548 = 0.0411. Exercise price on put used to price the chooser is 125(1.0456)-0.0411 = 124.77.
u Using Black-Scholes-Merton model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.
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Exotic Options (continued)
n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by
the sequence of prices followed by the asset and not just by the price of the asset at expiration.
u We shall price these options using a binomial framework. See Table 14.6 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed.
u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent
options. Monte Carlo simulation (see Appendix 14) is often used.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the
average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable as a measure of the underlying risk. Also useful for cases where market can be manipulated.
u See Table 14.7 for example of pricing Asian options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of
the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.
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Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. • lookback call: exercise price set at minimum price during option’s
life • lookback put: exercise price set at maximum price during option’s
life • fixed-strike lookback call: payoff based on maximum price during
option’s life (instead of final price) compared to fixed strike • fixed-strike lookback put: payoff based on minimum price during
option’s life (instead of final price) compared to fixed strike
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Exotic Options (continued)
n Path-Dependent Options (continued) u Lookback options (continued): F See Table 14.8 for example of pricing lookback options.
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Exotic Options (continued)
n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a
certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out-options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option.
F See Table 14.9 for example of pricing. F Barrier options are normally cheaper than ordinary options because
they provide payoffs for fewer outcomes than ordinary options.
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(Return to text slide)
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Exotic Options (continued)
n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow
options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options
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Chance/Brooks An Introduction to Derivatives and Risk
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Some Important New Derivatives
n Electricity Derivatives u Electricity is a non-storable asset u These derivatives are difficult to price
n Weather Derivatives u Measures of weather activity F Heating degree days and cooling degree days F Quantity of rain or snow F Financial loss caused by weather
u Pricing is difficult but not impossible; a lot of data are available on weather
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Summary
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Appendix 14: Monte Carlo Simulation n A method of using random numbers designed to simulate
the random observations of prices of an asset. The simulated series of asset prices at expiration is then converted to an equivalent series of option prices at expiration.
n Then the current option price is the discounted average of the option prices obtained at expiration from the simulation.
n Random prices can be simulated by drawing a standard normal random variable, e, and inserting into the formula
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Appendix 14: Monte Carlo Simulation (continued)
u where Dt is the length of the time interval over which the stock price change occurs.
u Note: simulating a standard normal random variable can be done approximately as the sum of twelve unit uniform random numbers (in Excel, “=Rand( )”) minus 6.0.
n Each simulated stock price is treated as the stock price at expiration; thus, Dt is the maturity in years of the option.
u For each simulated stock price, compute the option price at expiration using the intrinsic value.
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Appendix 14: Monte Carlo Simulation (continued)
u Take the average of all of the option prices at expiration. u Discount the average over the life of the option at the risk-free rate. This
is the estimate of the current option price.
n This procedure will probably require at least 50,000 random numbers for a standard option and more for exotic and complex options and derivatives.
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Some Important New Derivatives
n Electricity Derivatives u Electricity is a non-storable asset u These derivatives are difficult to price
n Weather Derivatives u Measures of weather activity F Heating degree days and cooling degree days F Quantity of rain or snow F Financial loss caused by weather
u Pricing is difficult but not impossible; a lot of data are available on weather
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Appendix 14: Monte Carlo Simulation n A method of using random numbers designed to simulate
the random observations of prices of an asset. The simulated series of asset prices at expiration is then converted to an equivalent series of option prices at expiration.
n Then the current option price is the discounted average of the option prices obtained at expiration from the simulation.
n Random prices can be simulated by drawing a standard normal random variable, e, and inserting into the formula
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Appendix 14: Monte Carlo Simulation (continued)
u where Dt is the length of the time interval over which the stock price change occurs.
u Note: simulating a standard normal random variable can be done approximately as the sum of twelve unit uniform random numbers (in Excel, “=Rand( )”) minus 6.0.
n Each simulated stock price is treated as the stock price at expiration; thus, Dt is the maturity in years of the option.
u For each simulated stock price, compute the option price at expiration using the intrinsic value.
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Appendix 14: Monte Carlo Simulation (continued)
u Take the average of all of the option prices at expiration. u Discount the average over the life of the option at the risk-free rate. This
is the estimate of the current option price.
n This procedure will probably require at least 50,000 random numbers for a standard option and more for exotic and complex options and derivatives.
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Some Important New Derivatives
n Electricity Derivatives u Electricity is a non-storable asset u These derivatives are difficult to price
n Weather Derivatives u Measures of weather activity F Heating degree days and cooling degree days F Quantity of rain or snow F Financial loss caused by weather
u Pricing is difficult but not impossible; a lot of data are available on weather
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Appendix 14: Monte Carlo Simulation n A method of using random numbers designed to simulate
the random observations of prices of an asset. The simulated series of asset prices at expiration is then converted to an equivalent series of option prices at expiration.
n Then the current option price is the discounted average of the option prices obtained at expiration from the simulation.
n Random prices can be simulated by drawing a standard normal random variable, e, and inserting into the formula
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Chance/Brooks An Introduction to Derivatives and Risk
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Appendix 14: Monte Carlo Simulation (continued)
u where Dt is the length of the time interval over which the stock price change occurs.
u Note: simulating a standard normal random variable can be done approximately as the sum of twelve unit uniform random numbers (in Excel, “=Rand( )”) minus 6.0.
n Each simulated stock price is treated as the stock price at expiration; thus, Dt is the maturity in years of the option.
u For each simulated stock price, compute the option price at expiration using the intrinsic value.
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Appendix 14: Monte Carlo Simulation (continued)
u Take the average of all of the option prices at expiration. u Discount the average over the life of the option at the risk-free rate. This
is the estimate of the current option price.
n This procedure will probably require at least 50,000 random numbers for a standard option and more for exotic and complex options and derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Some Important New Derivatives
n Electricity Derivatives u Electricity is a non-storable asset u These derivatives are difficult to price
n Weather Derivatives u Measures of weather activity F Heating degree days and cooling degree days F Quantity of rain or snow F Financial loss caused by weather
u Pricing is difficult but not impossible; a lot of data are available on weather
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Appendix 14: Monte Carlo Simulation n A method of using random numbers designed to simulate
the random observations of prices of an asset. The simulated series of asset prices at expiration is then converted to an equivalent series of option prices at expiration.
n Then the current option price is the discounted average of the option prices obtained at expiration from the simulation.
n Random prices can be simulated by drawing a standard normal random variable, e, and inserting into the formula
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Chance/Brooks An Introduction to Derivatives and Risk
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Appendix 14: Monte Carlo Simulation (continued)
u where Dt is the length of the time interval over which the stock price change occurs.
u Note: simulating a standard normal random variable can be done approximately as the sum of twelve unit uniform random numbers (in Excel, “=Rand( )”) minus 6.0.
n Each simulated stock price is treated as the stock price at expiration; thus, Dt is the maturity in years of the option.
u For each simulated stock price, compute the option price at expiration using the intrinsic value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Appendix 14: Monte Carlo Simulation (continued)
u Take the average of all of the option prices at expiration. u Discount the average over the life of the option at the risk-free rate. This
is the estimate of the current option price.
n This procedure will probably require at least 50,000 random numbers for a standard option and more for exotic and complex options and derivatives.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Some Important New Derivatives
n Electricity Derivatives u Electricity is a non-storable asset u These derivatives are difficult to price
n Weather Derivatives u Measures of weather activity F Heating degree days and cooling degree days F Quantity of rain or snow F Financial loss caused by weather
u Pricing is difficult but not impossible; a lot of data are available on weather
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Summary
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Appendix 14: Monte Carlo Simulation n A method of using random numbers designed to simulate
the random observations of prices of an asset. The simulated series of asset prices at expiration is then converted to an equivalent series of option prices at expiration.
n Then the current option price is the discounted average of the option prices obtained at expiration from the simulation.
n Random prices can be simulated by drawing a standard normal random variable, e, and inserting into the formula
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Appendix 14: Monte Carlo Simulation (continued)
u where Dt is the length of the time interval over which the stock price change occurs.
u Note: simulating a standard normal random variable can be done approximately as the sum of twelve unit uniform random numbers (in Excel, “=Rand( )”) minus 6.0.
n Each simulated stock price is treated as the stock price at expiration; thus, Dt is the maturity in years of the option.
u For each simulated stock price, compute the option price at expiration using the intrinsic value.
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Chance/Brooks An Introduction to Derivatives and Risk
Management, 9th ed. Ch. 14: 0
Appendix 14: Monte Carlo Simulation (continued)
u Take the average of all of the option prices at expiration. u Discount the average over the life of the option at the risk-free rate. This
is the estimate of the current option price.
n This procedure will probably require at least 50,000 random numbers for a standard option and more for exotic and complex options and derivatives.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.