Chapter 5 Options

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Chapter 5

Determination of Forward and

Futures Prices

Options, Futures, and Other Derivatives, 8th Edition,Copyright © John C. Hull 2012 1



OutlineHow forward prices and future prices are related tothe spot price.

Forward contracts are easier to analyze than futurescontracts because there is no daily settlement.

Forward and futures price of an asset are usuallyvery close when their maturities are the same.

Use arbitrage arguments to determine the forward &future prices of investment asset from its spot price

and observable market variables.

2



Consumption vs I nvestment AssetsInvestment assets are assets held by significantnumbers of people purely for investment purposes(Examples: stocks, bonds, gold, silver)

N.B: investment assets do not have to be heldexclusively for investment. Silver has a number ofindustrial uses. However, they do have to satisfythe requirement that they are held by significant

numbers of investors solely for investments.Consumption assets are assets held primarily forconsumption and not usually for investmentpurposes (Examples: copper, oil, corn)

3



Short Sell ingShort selling involves selling securities you do notown

Your broker borrows the securities from another client

and sells them in the market in the usual way. Consider short selling a stock:

• Shares are borrowed from a different owner/broker

• If the broker runs out of shares to borrow, the investor

must close out the position immediately (short-

squeezed)• All income from the stock (dividends, etc) is returned to

the stock’s owner.




Short Sell ing (continued)

At some stage you must buy thesecurities so they can be replaced in the

account of the client (close out theposition)

You must pay dividends and otherbenefits the owner of the securitiesreceives

There may be a small fee for borrowingthe securities




Margin AccountIn a short sale the investor is required tomaintain a margin account with the broker

It can consists of cash or marketablesecurities.




ExampleYou short 500 shares in April when the price is $120and close out the short position three months laterwhen the price is $100.

During the three months a dividend of $1 per share ispaid in May.

What is your profit?

Profit=500x120-500-500x100= $9,500

What would be your loss if you had bought 500shares?

Loss= $9,500 (mirror image cash flow of the shortsale)




Assumptions Assume the following holds for our major marketparticipants:

• Market participants are subject to no transaction costwhen they trade

• Market participants are subject to the same tax rate on

all net trading profits

• Market participants can borrow money at the samerisk-free rate of interest as they can lend money

• Market participants take advantage of arbitrage

opportunities as they occur.

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Notation for Valuing Futures and

Forward Contracts

9



Pricing forward contracts on investments

Asset wi th no income

Easiest forward contracts to price are non-income producing contracts. For example

non-dividend-paying stocks and zero-couponbonds.




Example Cont’d




General Rule for Nonincome

producing Futures

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Example




General Rule for I ncome

Producing Forwards cont’d




Example




Pricing forward contracts on

investments with known yield




Valuing a Forward Contract





By considering the difference between acontract with delivery price K and a contract

with delivery price F 0 we can deduce that:the value of a long forward contract, ƒ, is

( F 0 – K )e – rT

the value of a short forward contract is

( K – F 0 )e – rT




Example




Forward vs Futures PricesWhen the maturity and asset price are the same, forwardand futures prices are usually assumed to be equal.

(Eurodollar futures are an exception)When interest rates are uncertain they are, in theory,slightly different:

A strong positive correlation between interest rates and the assetprice implies the futures price is slightly higher than the forward

price A strong negative correlation implies the reverse




Futures Prices on Stock I ndices




Example cont’d




Remark




Futures and Forwards on

Currencies

A foreign currency is analogous to asecurity providing a yield

The yield is the foreign risk-free interestrate

It follows that if r f is the foreign risk-free

interest rate


F S e r r T f

0 0

( )



Explanation of the Relationship

Between Spot and Forward


1000 units offoreign currency

(time zero)

1000S 0 dollarsat time zero

1000S 0erT

dollars at time T

T

e T r f

timeat

currencyforeign

of units1000

T

e F T r f

timeatdollars

1000 0



Remarks on Foreign Currency

Futures




Futures on Commodities




Example

Options Futures and Other Derivatives 8th Edition

Chapter 5 Options

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