Financial Engineering 5

References: Villalobos, Luenberger, Faerber, Investopedia

Lecture 13

Introduction to Derivatives Part 1

Lecture Topics • Introduction to Derivative Securities • Swaps • Forwards • Examples

Derivatives • Derivatives are securities such as options and futures

contracts, whose value depends on the performance of an underlying asset such as a stock or contract.

• Some derivatives are classified by: – The type of underlying asset such as an equity, foreign

exchange, interest rate, etc. – The relationship with the underlying asset including options,

futures, and swaps. – The market which they are traded, such as an exchange,

over the counter (OTC), etc. – The derivative’s complexity including plain vanilla or exotic.

• Derivatives have contracts.

Why Use a Derivative? • Gain leverage, a small movement in the value of the underlying

asset can cause a large change in the value of the derivative.

• Making a profit (speculation) if the value of the underlying asset moves the way it is expected.

• Hedging (risk reduction) by taking positions on derivative contracts that moves in an opposite direction to the main position.

• Making a profit by getting a derivative position when it is not possible to get a position in the underlying asset, such as weather derivatives.

– Look up weather derivative in Wikipedia and Investopedia.

Over The Counter Market • Over The Counter (OTC) derivatives are contracts that are

traded (and privately negotiated) directly between two parties. • Products such as swaps, forward rate agreements, and exotic

options are almost always traded in the OTC market. • The OTC market consists of banks and other highly

sophisticated organizations such as hedge funds. • The OTC derivative market is the largest market for derivatives.

– The notional amount is approximately US$700 trillion. – Of this total notional amount, the contracts were related to:

78% Interest Rate 0.5% Commodity 5% Credit Default Swaps (CDS) 1% Equity 9% Foreign Exchange 6.5% Other Contracts

• Because OTC derivatives are not traded on an exchange, there is no central counter-party and they are a counter-party risk.

• This is the market where the recent mortgage problems arose. http://www.bis.org/statistics/derstats.htm

Exchange Market • A derivatives exchange is a market where individuals trade

standardized contracts that have been defined by the exchange. • Exchange Traded Derivative contracts (ETD) are traded in the

specified markets. • A derivatives exchange acts as an intermediary to all related

transactions; third party that can help reduce the risk. • According to the Bank of International Settlements (BIS), the

combined turnover in the world's derivatives exchanges totaled USD 5.3 trillion per day during 2013.

• Some types of derivative instruments may also trade on traditional exchanges.

• The world's largest derivatives exchanges based on the number of transactions are the Korea Exchange, Eurex, and Chicago Mercantile Exchange (CME Group).

Derivatives and Risk • Forward contracts, futures contracts, and options are the most

common types of derivatives.

• Derivatives are generally used by institutional investors to increase overall portfolio return or to hedge portfolio risk.

• We will focus on the use of derivative for risk management.

• The use of derivatives for risk management has attracted a lot of attention lately but it has a long history.

• The feudal lords of Japan in the 1600’s used a market called Cho-ai-mai to manage the volatility in the price of rice.

Forward Contracts • A forward contract is a non-standardized contract between two

parties to buy or sell an asset at a specified future time at a price agreed upon today.

• The forward contract is between two parties: the buyer and the seller.

– The buyer is said to be “long”, the seller is said to be “short”.

– Being long or short a given amount is the position of the party.

• The “forward price” is the price that applies at delivery. • The open market for immediate delivery of a commodity is

called the Spot Market. • The initial contract is usually set in such a way that the initial

payment for the contract is zero. • A key assumption in determining the price of the contracts is

arbitrage free.

Forward Contract Example • Forward Contract is a cash market transaction in which delivery of the

commodity is deferred until after a certain date specified in the contract.

– The price is determined at the initial trade date. • The Contract is agreed upon at time zero and settled at time N.

• Very often cash is delivered, instead of the commodity . • The amount of the payment is determined by the spot price of the

commodity at time n. • A concern might be: how do we determine the forward price at the time

the contract is signed?

0 1 2 3 N-1 N Cash

Commodity

. . . . .

Forward Contract Brewer Example • Suppose that a brewer needs 100 tons of special barley six

months from now to produce a batch of specialty beer sold during the Christmas holidays.

• Since the price of barley is highly unstable, the brewer wants to get a “long” position in a contract for 100 tons of barley to be delivered 6 months from today.

• What should be the forward price in $/ton for this contract?

• Assumptions: – The current price for barley is $200/ton. – The cost of storing a ton of barley is of $1/ton per month. – The risk free interest rates are given by the yield curve of the

US treasury securities.

Forward Prices • Define F as the forward price or the price agreed upon in the

contract to deliver a unit of the commodity. • Define f as the current value of the contract. • The forward price F is set such that f =0 (The value of the

contract is zero when it is signed). • Suppose that at time t = 0:

– Spot price for the underlying asset of a commodity is S. – A forward contract is being prepared for the delivery of the

asset at time T. – What should be the price F such that f =0?

• Two key observations: – The value of the forward contract is determined by the spot

price of the commodity. – The value of the contract can be used to lend or borrow

money at the normal market interest rates; the interest rates structure should apply.

Contract Value • An easy way to visualize the worth of a contract is to look at the

payoff graphic:

FSt −

Profit

From the perspective of the buyer (Long Position)

FSt −

Profit

From the perspective of the seller (Short Position)

Forward Prices • Suppose that you buy in the spot market a unit of a commodity

at price S and at the same time you enter in a contract to sell that unit at time T for a price F.

• Then the theoretical forward price F should meet:

Where d(0,T) is the discount factor calculated using the risk-free market interest rate.

• Thus the theoretical forward price would be:

( )d 0,S T F=

( )d 0, SF

T =

Forward Prices • The previous formula assumed that no storage costs were

incurred. • If there are storage costs, then:

• Where c(k) is the maintenance cost at period k.

• An equivalent formulation is:

( ) ( )

( )∑ −

=

+= 1

0 ,,0

M

k Mkd kc

Md SF

( ) ( ) ( )∑ −

=

−= 1

0 ,0,0

M

k kckdFMdS

Brewer Example • To simplify the analysis let’s assume that the yield for the 6 month

US Treasury is the nominal monthly discount rate. – That is, the monthly interest rate = (0.14%/12)= 0.012% per

month. – Then the forward price of barley in the contract should be:

• Exercise: Suppose that brewer found a counterparty for a forward contract and the contract was signed. Two months later the spot price of the ton of barley increased to $205/ton. What is the value of the contract then?

( ) ( )

( )

( )

( )

( )

1

0

1

0 6

0, ,

100 120,000 20,614.181 1 1 0.00012 1 0.00012

M

k

M

k k

c kSF d M d k M

−

=

−

=

= + =

× = + =

+ +

∑

∑

Swaps • Swaps are financial products that are used to alter the exposure

of investment portfolios, or any series of cash flows. • The most common kind of swap is an interest rate swap. • In an interest rate swap, two parties agree to exchange periodic

interest payments based on a predetermined notional principal amount.

• In a typical interest rate swap one party will pay a fixed interest rate, while the other party agrees to pay a floating rate.

• For example, two parties may enter into an interest rate swap in which they agree to exchange interest payments on $100 million notional principal.

– In this swap, one counterparty may agree to pay a fixed rate of 7%.

– The other counterparty may agree to pay 3 month , London Interbank Offered Rate (LIBOR).

Swaps • The value of an interest rate swap changes as the level of

interest rates change.

• For instance, a fixed rate payer essentially has a fixed rate liability and a floating rate asset.

• If interest rates fell, the fixed rate payer would still have to pay the higher fixed rate.

• If the short-term rate received remained the same, the marked to market value of the fixed rate payer's position would be negative.

• Conversely, the fixed rate receiver would have a positive market to market position if the opposite occurred.

Interest Rate Swap • The most common and simplest swap is a "plain vanilla"

interest rate swap. • In this swap, Party A agrees to pay Party B a predetermined,

fixed rate of interest on a notional principal on specific dates for a specified period of time.

• Concurrently, Party B agrees to make payments based on a floating interest rate to Party A on that same notional principal on the same specified dates for the same specified time period.

• In a plain vanilla swap, the two cash flows are paid in the same currency.

– The specified payment dates are called settlement dates. – The time between are called settlement periods.

• Because swaps are customized contracts, interest payments may be made annually, quarterly, monthly, or at any other interval determined by the parties.

• Note, you are only paying the difference at the settlement dates.

Interest Rate Swap • For example, on December 31, 2010, Company A and Company

B enter into a four-year swap with the following terms: – Company A pays Company B an amount equal to 6% per

annum on a notional principal of $20 million. – Company B pays Company A an amount equal to one-year

LIBOR + 1% per annum on a notional principal of $20 million.

Example • LIBOR, or London Interbank Offer Rate, is the interest rate offered by

London banks on deposits made by other banks in the eurodollar markets. • The market for interest rate swaps frequently uses LIBOR as the base for

the floating rate. • For simplicity, let's assume the two parties exchange payments annually

on December 31, beginning in 2011 and concluding in 2015. • At the end of 2011, Company A paid Company B $20,000,000 x 6% =

$1,200,000. • On December 31, 2010, one-year LIBOR was 5.33%; therefore, Company B

will pay Company A $20,000,000 x (5.33% + 1%) = $1,266,000. • In a plain vanilla interest rate swap, the floating rate is usually determined

at the beginning of the settlement period. • Normally, swap contracts allow for payments to be netted against each

other. – Here, Company B pays $66,000, and Company A pays nothing. – At no point does the principal change hands, which is why it is referred

to as a "notional" amount.

Why Use a Swap? • The motivations fall into two basic categories: commercial needs and

comparative advantage. • The normal business operations of some firms lead to certain types of

interest rate or currency exposures that swaps can reduce. • For example, consider a bank, which pays a floating rate of interest on

deposits (i.e., liabilities) and earns a fixed rate of interest on loans (i.e., assets).

– The bank could use a fixed-pay swap (pay a fixed rate and receive a floating rate) to convert its fixed-rate assets into floating-rate assets, which would match up well with its floating-rate liabilities.

• Some companies have a comparative advantage in acquiring certain types of financing.

• A company may acquire the financing for which it has a comparative advantage, then use a swap to convert it to the desired type of financing.

• For example, consider a well-known U.S. firm that wants to expand its operations into Europe, where it is not well known.

• It will likely receive more favorable financing terms in the US; by using a currency swap, the firm ends with the Euros it needs to fund its expansion.

Investopedia

Assignments • Finish reading Luenberger Chapter 10. • Check out definitions for derivatives, forwards and swaps in

Investopedia and Wikipedia. • Make progress on your Literature Review!