Financial Engineering 5

References: Villalobos, Luenberger, Faerber, Investopedia

Lecture 14

Introduction to Derivatives Part 2

Lecture Topics • Introduction to Derivative Securities • Commodity Swaps • Currency Swaps • More 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.

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.

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.

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 market 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.

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

Commodity Swap Value • Commodities are physical assets such as metals, energy and agriculture. • A commodity swap is an agreement whereby a floating (or market or spot)

price is exchanged for a fixed price over a specified period. • Then in return, the counterparty would get payments based on the market

price for the commodity involved. • Value of a Commodity Swap

– Consider an agreement where party A receives spot market price for N units of a commodity for each period while paying a fixed amount X per unit for N units.

– If the agreement is made for M periods, the net cash flow stream received by party A is (S1 – X, S2 – X,…, SM – X ) multiplied by the number of units N, where Si denotes the spot price of the commodity at time i.

• The value of this stream is: ( )( )NXFidV i

M

i −=∑

=1 ,0

Where Fi is the forward price of one unit of the commodity at time i.

Example • Determine the fixed price for a fixed-for-floating gold price

swap, assuming settlement will be every six months, beginning 2 month months from today.

• The term will be 20 months and the floating price will be the spot price of gold on each settlement date.

• Gold futures prices and rates happened to be:

Time Gold Price Spot rates 2 310 r(0,0.16)=3.0% 8 316 r(0,0.66)=3.2%

14 320 r(0,1.16)=3.5% 20 324 r(0,1.66)=3.8%

Example • The Present value of the gold floating payments is:

PV = $1,230

• The present value of the agreement to sell gold at the fixed price Pfixed is:

• Pfixed = $317.378

• The trader will sell or buy a contract of gold at $317.378.

( ) ( ) ( ) ( )0.16 0.66 1.16 1.66 310 316 320 324

1 0.030 1 0.032 1 0.035 1 0.038 PV = + + +

+ + + +

( ) ( ) ( ) ( )0.16 0.66 1.16 1.66 1 1 1 1

1 0.030 1 0.032 1 0.035 1 0.038 fixed PV P

  = + + + 

+ + + +  

Example • Suppose that a company (ACME T Division) is in the business of

making ketchup and needs 20 tons of fresh tomatoes per month. • The sale price of ketchup can be considered fixed; however, the price

of fresh tomatoes is highly variable as evidenced by the graph below (Average price = $0.678/#, Std. dev. = 0.157) .

• Suppose that the Company wants to enter into a commodity swap for 10 tons per month for the next 3 years; what should be the fixed price (x) that the Company should be willing to pay in such a way that the current value of the contract is zero?

0

5

10

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40

3/ 3/

20 08

4/ 3/

20 08

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10 /3

/2 00

8 11

/3 /2

00 8

12 /3

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3/ 20

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/3 /2

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9 12

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20 10

3/ 3/

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4/ 3/

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10 /3

/2 01

0 11

/3 /2

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12 /3

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3/ 20

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3/ 20

11

Price in $ Roma Tomatoes Dallas 25# case

Example • Assumptions:

– Prices follow a normal distribution with a mean of $0.657/# and stdev = 0.1431; (we’ll address the validity of this in a later lecture)

– We will use the 3 year T Bond as the discount rate, 1.17%.

• Using the formula:

• By using the mean as the forward price we can see that the fixed price should be the same as the forward price to obtain a value of the contract of zero at the onset.

( )( )NXFidV i M

i −=∑

=1 ,0

Example • Now assume that the monthly tomato prices will behave according

to the relationship:

where s0 is the price per pound of tomatoes at month 0 (today), st is the price at t months from now, and we assume that s0 = $0.657/#.

• What should be the fixed price so that the price of the contract at the onset is zero?

• Suppose that at the time of the swap settlement, 10 months after the contract is signed, the price per pound of tomatoes changes to:

$1/# $0.5/#

• How much is the value of contract? • What is the amount of the next swap settlement?

t

t ess   

   

 +−

= 2 1331.0005.0

0

2

Value of an Interest Swap • Consider an agreement where party A agrees to make payments

of a fixed rate r of interest on a notional principal from M periods.

• The cash flow stream for party A is:

(C0 – r, C1 – r, . . . . . , CM – r ) times the principle N

• The Ci denotes the floating rates.

( ) ( ) NidrMdV M

i  

  

 −−= ∑

=1 ,0,01

• The value of the swap is:

Luenberger Exercise • Suppose that the current term structure of interest rates is

(0.070, 0.073, 0.077, 0.081, 0.084, 0.088).

• A plain vanilla interest rate swap will make payments at the end of each year equal to the floating short rate that was posted at the beginning of that year.

• A six year swap having a notional principal of $10,000,000 is being considered.

• What is the value of the floating rate portion of the swap?

• What rates of interest for the fixed portion of the swap would make the two sides of the swap equal?

Luenberger Exercise 10.11 • The value of the variable part of the swap as implied by the term

structure is:

• The value of the fixed part of the swap is:

• Setting both value equal to each other and solving for r we find that r = 8.62% (Luenberger shows 8.64% due to rounding differences.

( ) ( )6

11 0, 1 10,000,000 3,971,259 1 0.088float

V d M N  

= − = − =     +  

( )

( ) 1

0,

0.9346 0.8685 0.8005 0.7323 0.6681+0.6029 10,000,000

46,069,319

M

fixed i

V r d i N

r

r

=

 =    = + + + +   =

∑

Valuing a Swap After Origination • If prices or rates subsequently change, the value of the swap

will change.

• It will become of positive value for one party, and negative value for the other (the party who could default).

• The calculations remain the same. – You have to compute the PV of the fixed cash flows and of

the expected variable cash flows

• It is important to note that you can decide if the PV of the floating Cash Flow = Notion Principal immediately after a Cash Flow has been swapped.

Currency Swaps • Currency swaps are foreign exchange agreements between two

parties to exchange aspects a loan in one currency for equivalent aspects of an equal loan in another currency.

• Currency swaps are Over The Counter derivatives, and are closely related to interest rate swaps.

• In a currency swap, a principal must be specified in each currency and the principal amounts are exchanged at the beginning and end of the life of the swap.

• The principal amounts are approximately equal, given the exchange rate at the beginning of the swap.

Example • Company X pays a fixed rate of 7% in dollars and receives a

fixed interest rate of 9% in euros. • Interest payments are made once per year and the principal

amounts are $25 million and ∈20 million.

Date Dollar Cash Flow Millions

Euro Cash Flow Millions

6/5/2006 +25.00 -20.00

6/5/2007 -1.75 +1.8

6/5/2008 -1.75 +1.8

6/5/2009 -1.75 +1.8

6/5/2010 -1.75 +1.8

6/5/2011 -26.75 +21.8

Currency Swap Exercise • Consider the following currency swap:

• An agreement to receive 2% on a UK Sterling principal of 1,045 million pounds and pay 6% on a US dollar principal of $10 million every year for a swap tenor of 5 years.

• Suppose that 3 months after the swap origination, the UK interest rate rises to 3%, but US interest rates remain the same.

• Solve the exercise.

Another Forward Contract Example • Suppose that you have the following information on a bond:

Bond Coupon Maturity Date YTM Price FED HOME LN BKS 4.5 2/15/21 1.604 105.65

• Today is February 24. • Using an annual interest rate of 4% determine the forward price

for this bond a year from now. • Two coupon payments of $22.5 will place on 8/15 and 2/15.

• F = 1054.11

( ) 193 090.04 0.04

0.04 365 3651056.5 22.5 22.5F e e e            

    = − −      

   

Arbitrage Free Forward Prices • Forward prices are such that they preclude arbitration.

• Suppose that in the previous case the forward price of the bond had been quoted as $1200.00.

• In this case you can make money by: – Sell the bond forward at $1200. – Borrow $1056.50 at 4%. – Buy one bond at $1056.50 – Get the coupon proceedings. – Deliver the bond a year later using the proceedings to repay

the loan. – Profit = 1200 – 1056.5(1.04) = 10.124 plus the coupons!

Arbitrage Free Forward Prices

• Now suppose that the dealer quotes you the bond at $1000. – Buy the bond forward at $1000. – Borrow one bond at the current price. – Sell the bond in the spot market at $1056.5. – Deposit $1056.5 at 4%. – Cash the deposit at (105.65)(1.04)=$1098.7. – After one year receive the bond at 1000. – Profit = 1098.7 - 1000

Forward Foreign Exchange (FX) • The contract holders are obligated to buy or sell the currency at

a specified price, at a specified quantity and on a specified future date.

• These contracts cannot be transferred.

• Assume that the current exchange (spot) rate is 1.5 Dollars per Euro and 10.75 Mexican pesos per dollar.

– Also assume that you enter into two independent forward FX contracts to deliver 6,666,666.67 Euros and 10,750,000 Mexican pesos a year from now.

• What is the forward exchange to make the current price of these contracts zero if we know that the current risk-free rates for US dollars is 3.84%, that for Euros is 4.3% and for Mexican pesos is 7.2%?

Short Position Perspective (Contract Issuer) EurosDollars

t = now

t = one year from now -6,666,667

+6,391,818

Amount delivered in

a year (principal + Interests)

Amount deposited in a

risk free account

-9,587,728

Amount lent at risk free rate

and converted to Euros

-9,955,897

The forward conversion rate must be 1.4934 Euros per dollar to have a risk free, arbitrage free transaction

-6,391,818

+6,666,667

+9,955,897

3.84% 4.3%

Different Scenarios • Suppose that a year from now the exchange rate is 1.7 Euros per

dollar • The value of the contract is (1.7 – 1.493)6,666,667 = -1,377,437 dollars. • However we will receive 6,666,667 euros at 1.493 which is exactly what

we need to cover our loan (9,955,896) • Suppose that the rate is 1.1% • In this case the value of the contract at maturity is 2,622,563. • However we will receive 6,666,667 euros at 1.493 which is exactly what

we need to cover our loan (9,955,896) • No risk, no arbitrage price.

• As an exercise determine the forward rate for the Mexican peso.

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

Investopedia and Wikipedia.