Determination of Forward

and Futures Prices

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

A popular trading (arbitrage) strategy is the short-

selling or shorting.

This trade involves selling an asset that is not owned

by the seller, and that the seller has borrowed.

Short selling is motivated by the belief that an asset’s

price will decline, enabling it to be bought back at a

lower price to make a profit.

Short selling is largely prompted by speculation.

Short sellers take advantage of any likely arbitrage

opportunities.

2

Short selling

Investor A instructs a broker to short 500 shares who

does not own.

The broker will borrow the shares from another client,

say Investor B, and sell them in the market.

Investor A can maintain the short position for as long

as desired, provided there are always shares to

borrow.

At some stage, Investor A will close out the position by

purchasing the 500 shares.

The 500 shares are replaced in the account of Investor

B.

3

Short selling

Investor A takes a profit if the stock price has declined

and a loss if it has risen.

If at any time while the contract is open the broker is

not able to borrow shares, investor A is forced to close

out the position, even if not ready to do so.

Investor A must pay dividends and other benefits to the

owner of the shares, i.e. to Investor B via the broker.

A broker fee is also charged to Investor A for borrowing

the shares.

4

Short selling

Investor A is required to maintain a margin account

with the broker.

The margin account consists of cash or marketable

securities deposited by the Investor with the broker to

guarantee that the investor will not walk away from the

short position if the share price increases.

An initial margin is required and if there are adverse

movements (i.e., increases) in the price of the asset

that is being shorted, additional margin may be

required.

If the additional margin is not provided, the short

position is closed out.

5

Short selling

Consider the position of Investor A who shorts 500

shares in April when the price per share is $120.

Investor A closes out the position by buying them back

in July when the price per share is $100.

Suppose that a dividend of $1 per share is paid in May.

Investor A receives 500 x $120 = $60,000 in April when

the short position is initiated.

The dividend leads to a payment by Investor A of 500

x $1 = $500 in May.

Investor A also pays 500 x $100 = $50,000 for shares

when the position is closed out in July.

The net gain is: $60,000 - $500 - $50,000 = $9,500

assuming there is no fee for borrowing the shares. 6

Short selling: Example

Massive short selling is a practice that is often

observed after the burst of a price bubble.

Examples are the Dutch Tulip-mania in the

seventeenth century, the U.S. stock price crash in

1929, the NASDAQ price bubble of 1998-2000, etc.

Since the practice of short selling is alleged to magnify

the decline of asset prices, it has been banned and

restricted many times during history.

As such, short selling bans have been commonly used

as a regulatory measure to stabilise prices during

downturns in the economy.

7

Short selling

The most recent example was in September of 2008 with

the prohibition of short selling by the U.S. Securities and

Exchange Commission (SEC) for 799 financial companies in

an effort to stabilise those companies.

At the same time the U.K. Financial Services Authority

(FSA) prohibited short selling for 32 financial companies.

On September 22, Australia enacted even more extensive

measures with a total ban of short selling.

8

Short selling

S0: Spot price today

F0: Futures or forward price today

T: Time until delivery date

r: Risk-free interest rate for maturity T

r is the rate at which money is borrowed or lent when

there is no credit risk, so that the money is certain to be

repaid.

Participants in derivatives markets have traditionally

assumed that LIBOR rates are the relevant risk-free

rates.

9

Pricing Futures and Forward Contracts

LIBOR is a reference interest rate; it is designed to reflect

the rate of interest at which banks are prepared to make

large wholesale deposits with other banks.

LIBOR is quoted in all major currencies for maturities up to

12 months:

E.g., 1-month LIBOR is the rate at which 1-month deposits

are offered.

A deposit with a bank can be regarded as a loan to that

bank.

A bank must satisfy certain creditworthiness criteria in order

to be able to receive deposits from another bank at LIBOR.

Typically, it must have a AA credit rating.

LIBID is the rate which a AA bank is prepared to pay on

deposits from another bank

10

Pricing Futures and Forward Contracts

Arbitrage arguments are used to determine the forward

and futures prices of an asset from its spot price and

other observable market variables.

In other words, it is the trading activities of market

participants to take advantage of arbitrage

opportunities as they occur that determine the

relationship between forward and spot prices.

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Pricing Futures and Forward Contracts

Suppose an asset that provides the holder with no

income: non-dividend-paying stocks, or zero-coupon

bonds.

Consider a long forward contract to purchase a non-

dividend-paying stock in 3 months.

Assume the current stock price is $40 and the 3-month

risk-free interest rate is 5% per annum.

Suppose that the forward price is relatively high at $43.

An arbitrageur can borrow $40 at the risk-free interest

rate of 5% per annum, buy one share, and short a

forward contract to sell one share in 3 months.

12

Forward price of an asset

13

Forward price of an asset

If the spot price of an investment asset is S0 and the

futures price for a contract deliverable in T years is F0,

then:

F0 = S0erT

where r is the T-year risk-free rate of interest.

14

Interest Rates – Compounding

When we compound m times per year at rate R an

amount A grows to A(1+R/m)m in one year

Compounding frequency Value of $100 in one year at 10%

Annual (m=1) 110.00

Semiannual (m=2) 110.25

Quarterly (m=4) 110.38

Monthly (m=12) 110.47

Weekly (m=52) 110.51

Daily (m=365) 110.52

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Interest Rates – Continuous Compounding

In the limit as we compound more and more frequently

we obtain continuously compounded interest rates.

$100 grows to $100eRT when invested at a

continuously compounded rate R for time T.

$100 received at time T discounts to $100e-RT at time

zero when the continuously compounded discount rate

is R.

At the end of the 3 months, the arbitrageur delivers the

share and receives $43.00

The sum of money required to pay off the loan is:

Hence, the arbitrageur locks in a profit of $43.00 -

$40.50 = $2.50 at the end of the 3-month period.

16

Forward price of an asset

Suppose that the forward price is at $39.00

An arbitrageur can short one share, invest the money

brought in from a transaction of the short sale at 5%

per annum for 3 months, and take a long position in a

3-month forward contract.

The proceeds of the short sale in 3 months grow to:

At the end of the 3 months, the arbitrageur pays $39,

takes delivery of the share under the terms of the

forward contract, and uses it to close out the short

position.

A net gain of $40.50 - $39.00 = $1.50 is made at the

end of the 3 months. 17

Forward price of an asset

A net gain of $40.50 - $39.00 = $1.50 is made at the

end of the 3 months.

Overall:

a) The first arbitrage works when the forward price is

greater than $40.50.

b) The second arbitrage works when the forward price is

less than $40.50.

We deduce that for there to be no arbitrage the forward

price must be exactly $40.50.

18

Forward price of an asset

Short sales are not possible for all assets and

sometimes is banned from authorities.

This does not matter: we can still derive and use the

same formula (F0 = S0erT ).

All that we require is that there be a significant number

of people who hold the asset purely for investment

(and by definition this is always true of an investment

asset).

If the forward price is too low, they will find it attractive

to sell the asset and take a long position in a forward

contract.

19

Forward price of an asset

Consider a forward contract on an asset that will

provide a perfectly predictable cash income to the

holder.

Examples are stocks paying known dividends and

coupon-bearing bonds.

We adopt the same approach as before and obtain:

F0 = (S0 – I )erT

where I is the present value of the income during life of

forward contract.

20

A Known Income

Consider the situation where the asset underlying a

forward contract provides a known yield.

This means that the income is known when expressed

as a percentage of the asset’s price at the time the

income is paid.

Suppose that an asset is expected to provide a yield of

5% per annum.

This could mean that income is paid once a year and is

equal to 5% of the asset price at the time it is paid.

The formula is: F0 = S0 e(r–q )T where q is the average

yield during the life of the contract (expressed with

continuous compounding).

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A Known Yield

When the short-term risk-free interest rate is constant,

the forward price for a contract with a certain delivery

date is in theory the same as the futures price for a

contract with that delivery date.

When interest rates vary, forward and futures prices

are in theory no longer the same.

We can get a sense of the nature of the relationship by

considering the situation where the price of the

underlying asset, S, is strongly positively correlated

with interest rates.

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Forward Prices and Futures Prices

When S increases, an investor who holds a long

futures position makes an immediate gain because of

the daily settlement procedure.

The positive correlation indicates that it is likely that

interest rates have also increased.

Similarly, when S decreases, the investor will incur an

immediate loss.

23

Forward Prices and Futures Prices

An investor holding a forward contract rather than a

futures contract is not affected in this way by interest

rate movements.

A long futures contract will be slightly more attractive

than a similar long forward contract.

Hence, when S is strongly positively correlated with

interest rates, futures prices will tend to be slightly

higher than forward prices.

When S is strongly negatively correlated with interest

rates, a similar argument shows that forward prices will

tend to be slightly higher than futures prices.

24

Forward Prices and Futures Prices

We have introduced futures on stock indices and

showed how a stock index futures contract is a useful

tool in managing equity portfolios.

We now consider how index futures prices are

determined.

A stock index can usually be regarded as the price of

an investment asset that pays dividends.

The investment asset is the portfolio of stocks

underlying the index, and the dividends paid by the

investment asset are the dividends that would be

received by the holder of this portfolio.

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Futures Prices of Stock Indices

It is usually assumed that the dividends provide a

known yield rather than a known cash income.

The following formula gives the futures price of stock indices:

F0 = S0 e(r–q )T

where q is the average dividend yield on the portfolio represented by the index during life of contract.

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Futures Prices of Stock Indices

When F0 > S0e(r-q)T an arbitrageur buys the stocks

underlying the index and sells futures.

When F0 < S0e(r-q)T an arbitrageur buys futures and

shorts or sells the stocks underlying the index.

To do index arbitrage, a trader must be able to trade

both the index futures contract and the portfolio of

stocks underlying the index very quickly at the prices

quoted in the market.

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Index Arbitrage

28

Forward and Futures Prices: Summary