chap2

CHAPTER 2. FUTURES MARKET AND PRICES 20

Chapter 2

Futures market and prices

2.1 Warming up

2.1.1 What is a risk-free interest rate?

Let us consider the following four portfolios of securities:

1. A portfolio of Treasury bills, i.e., United States government debt securities

maturing in less than 1 year

2. A portfolio of long-term United States government bonds

3. A portfolio of long-term corporate bonds

4. S&P500, i.e., Standard and Poor’s Composite Index, which represents a

portfolio of common stocks of 500 large firms

The portfolios offer different degrees of risk. Treasury bills are about as safe

as an investment as you can make. There is no risk of default, and their short

maturity means that the prices of Treasury bills are relatively stable. In fact,

an investor who wishes to lend money for, say, 3 months can achieve a perfectly

certain payoff by purchasing a Treasury bill maturing in 3 months.

By switching to long-term government bonds, the investor acquires an asset

whose prices fluctuate as interest rates vary. An investor who switches from


Table 2.1: Average rates of return (ARR) and Average risk premium, 1926-1988

(figures in % per year).

Portfolio Average ARR Average Risk Premium

Treasury bills 3.6 0

Government bonds 4.7 1.1

Corporate bonds 5.3 1.7

S&P500 12.1 8.4

government to corporate bonds accepts an additional default risk. An investor

who shifts from corporate bonds to stocks has a direct share in the risks of the

enterprise.

In the study of financial mathematics, we normally assume that we can borrow

and save money at the risk-free rate of interest.1 This assumption is not valid for

an individual but for a large investment house.

2.1.2 Continuous compounding

For the study of pricing derivatives, the continuously compound interest rates

are used.

There are two types of interests: simple and compound. Interest is said to be

simple if it is withdrawn at the end of each interest period so that the value of

the investment remains the same no matter how long the period for which it is

1What is the risk-free interest rate? Is it the long-term government bond rate, the London

interbank rate or the Bank of England base rate? Well, it may depend on the context. Any of

these rates may be considered to be the risk-free rate and, in fact, they differ only very slightly.

Another question is ’Is the risk-free rate really risk-free and does not change at all?’. The Bank

of England base rate changed from 6% at the begining of 2001 to 4% at the end of the year.

However, in this module we do not really worry about these problems. Despite the problems,

it is reasonably fair to assume for the study of options and forwards that the risk-free interest

rate is stable for the life time of these derivatives which is relatively short.


invested. With compound interest, on the other hand, the interest is added to

the investment at the end of each period (is compounded) so that the value of the

investment increases steadily.

Assume the interest rate per annum is R and the value of the initial investment

is So. If the rate is compounded m times a year then after n years, the value Sm

of the investment is

Sm = So

(

1 +R

m

)mn

. (2.1)

If it is compounded continuously, the value Sc of the investment is

Sc = So limm→∞

(

1 +R

m

)mn

. (2.2)

Using the definition of the exponent, we find

Sc = SoeRn. (2.3)

We can easily prove that(

1 +R

m

)mn

< eRn. (2.4)

To find the compatibility of the annual interest rates, suppose that Rc is a rate

of interest with continuous compounding and Rm is the equivalent rate with

compounding m times per annum. Assuming Sc = Sm, we get

SoeRcn = So

(

1 +Rm

m

)mn

(2.5)

which gives the following relation

Rc = m ln

(

1 +Rm

m

)

(2.6)

and

Rm = m(eRc/m − 1). (2.7)

In the world of finance, R is normally an annual rate and T is counted in year.

For example, the initial investment of So becomes

S = Soe0.05×0.25 (2.8)

by the three-month risk-free investment at the interest rate of 5% per annum.


2.2 What is a bond?

When you own a bond, you receive a fixed set of cash payoffs. Each year until

the bond matures, you get an interest payment and then at maturity you also

get back the face value of the bond. The face value of the bond is known as the

principal. The interest payment is called coupon payment. Bond investors would

say, for example, that this bond has a 5% coupon.

Sometimes bonds are sold at a discount on their face value, so that investors

receive a significant part of their return in the form of capital appreciation. The

ultimate is the zero coupon bond, which pays no interest at all; in this case all

the return consists of price appreciation.

2.3 Forward pricing

We consider forward pricing for there to be no arbitrage opportunities.

2.3.1 On an asset without income/cost

Examples: Non-dividend-paying stocks and zero coupon bonds.

The price G of the forward on an asset whose spot price is S must be

G = Ser(T−t) (2.9)

where r is the risk-free interest rate, T the time for the forward contract to mature

and t the current time. The duration of time T − t is counted by years.

If G > Ser(T−t), profits can be made by buying the underlying stocks (this is

financed by borrowing S pounds at the risk-free rate) and shorting forward con-

tracts. If G < Ser(T−t), profits can be made by shorting or selling the underlying

stocks and taking a long position in forward contracts. However, the arbitrage

chance does not last long because, for example, if G > Ser(T−t) everybody will

rush to buy the underlying stock and S will increase in due course.

We consider the following two portfolios for formal arguments (G: delivery

price, f : value of a long forward contract):


• Portfolio A: a long position of one unit of forward contract and an amount

of cash equal to Ge−r(T−t)

• Portfolio B: a long position of one unit of the underlying asset.

It is easily seen that the two portfolios have the same value. Thus

f + Ge−r(T−t) = S. (2.10)

We know that when the forward contract is to enter, its value is zero and the

delivery price is the same as the forward price. Otherwise there should be an

arbitrage chance. Taking f = 0, Eq.(2.9) is recovered:

Ge−r(T−t) = S. (2.11)

G is the price of the forward and f is its value.

2.3.2 On an asset which provides known cash income

Examples: Stocks paying known dividends and coupon-bearing bonds.

Assuming I as the present value of income to be received during the life of the

forward contract, the relationship between G and S is

G = (S − I)er(T−t). (2.12)

Example

Consider a 9-month forward contract on a stock with a price of £30. We

know that dividends of 50p and 80p per share will be paid after 3 months and 6

months respectively. We assume that the risk-free compound interest rate is 6%

per annum. The present value of the dividends is

I = 0.50 × e−0.06×0.25 + 0.80 × e−0.06×0.5 = 1.268.

The forward price is

G = (30 − 1.268)e0.06×0.75 = 30.05(pounds).


2.3.3 On an asset with continuous income

Example: Index portfolio

Assuming a continuous income at the rate q per annum, S − I in Eq.(2.12) is

the same as Se−q(T−t) and

G = Se(r−q)(T−t). (2.13)

2.3.4 On an asset with a cost to store

Examples: gold, vegetables and grains

If the present value of the storage cost is U , the spot price S has to be substi-

tuted by S + U in Eq.(2.9). If the storage cost per unit is a constant proportion,

u, of the spot price, the spot price S is substituted by Seu(T−t).

G = (S + U)er(T−t) ; G = Se(r+u)(T−t). (2.14)

NB: For some commodities, the forward price is not determined by the equation

above. Even when G < Se(r+u)(T−t), some asset holders are reluctant to short

the asset when they do not hold them for investment reasons.

As an example, consider a one-year forward contract on gold. It costs $1

semiannually per ounce to store gold. Suppose the payment is made in advance

every 0.5 year. Assume that the spot price is $450 and the risk-free rate is 7%

per annum for all maturities. What is the forward price?

U = 1 + 1 × e−0.07×0.5 = 1.9656

and the futures price

G = (450 + 1.9656)e0.07 = 487.74.

When G > (S + U)er(T−t), an arbitrage profit is earned by borrowing S + U

at the risk-free rate to purchase the commodity and shorting a forward on the

commodity. When G < (S + U)er(T−t), an arbitrage profit is earned by selling

the commodity at S + U and longing a forward. But who wants to buy the


commodity at S + U when its price is S? So, sell the commodity at S and save U

and long a forward to get a profit. More precisely, G ≤ (S + U)er(T−t) because

there are some owners who keep the commodity due to its consumption value.

2.3.5 Interest rate parity relation

Example: Foreign currencies

Assume that the value of the underlying asset S is the current price in the

domestic currency of one unit of the foreign currency. The foreign currency

considered as an asset has the property that the holder of the asset can earn

interest at the foreign risk-free interest rate rf . The price of a forward contract

on a foreign currency is given by

G = Se(r−rf )(T−t). (2.15)

For example, the risk-free interest rate in the UK is 6% per annum while

in the US is 4%. The US$ forward price for one-year delivery is $amount

×e0.02 =$amount×1.02.

2.3.6 Cost of carry

The relationship between forward prices and spot prices can be summarised in

terms of the cost of carry, c:

G = Sec(T−t). (2.16)

2.4 Marking to market the account

Forward contracts and futures are much alike, except that the former are private

contract between two parties and the latter are traded over-the-counter. To

minimise the contract defaults, an investor who buys a futures is requested to

deposit funds in a margin account. The margin account is there to safeguard

against the possibility of default.


The following is the contract specification for the corn futures at CBOT (Chicago

Board of Trade) on 15 January 2004.

$ Corn Contract

Maintenance Margin: $ 400 per contract

Initial Margin: $ 540 (initial margin mark up percentage = 135%)

Contract Size: 5,000 bushels

Deliverable Grades: No. 2 Yellow at par and substitutions at differentials

established by the exchange

Delivery Months: March, May, July, September, December

Minimum Tick (Size/Value): 1/4 cent per bushel ($12.50/contract)

Daily Price Limit: 20 cents/bu ($1,000/contract) above or below the previous

day’s settlement price.

Last Trading Day: The business day prior to the 15th calendar day of the

contract month.

Last Delivery Day: Second business day following the last trading day of the

delivery month.

Let us consider the following transaction in the CBOT.

Day 0 A customer longs two corn futures contracts (March delivery) at the clos-

ing price of $2.07 per bushel. Initial margin: 2 × $540 = $1, 080

Day 1 Closing futures price of day 1: $2.05 per bushel

Change in futures price: $2.05 - $2.07 = -$0.02 per bushel

Investor’s loss: 2 × 5, 000 × (−$0.02) = −$200

Margin Account Balance: $1, 080 − $200 = $880

Day 2 The price of the corn futures falls to $2.04 per bushel.

Change in futures price: $2.04 - $2.05 = -$0.01 per bushel

Investor’s loss: 2 × 5, 000 × (−$0.01) = −$100

Margin Account Balance: $880 − $100 = $780.

$780 is less than the maintenance margin by $20 so the customer receives a


margin call. He will have to pay the variation margin $1, 080 − $780 = $300. If

not, the customer’s position is closed.

Day 3 The price of the corn futures rises to $2.08 per bushel.

Change in futures price: $2.08 - $2.04 = $0.04 per bushel

Investor’s gain: 2 × 5, 000 × $0.04 = $400

Margin Account Balance: $1, 080 + $400 = $1, 480

$400 is more than the initial margin so the customer can take out $400.

The effect of the marking to market is that a futures contract is settled daily

rather than all at the end of its life.

It is straightforward to prove that the margin account payment becomes the

futures payment at maturity without considering an interest on the margin ac-

count.

proof

Denote the initial margin by I, the maintenance margin by M and the futures

price on day i by Fi. On day 0, the initial margin I is paid into the margin

account. On day 1, I + (F1 − F0) is in the margin account. On day 2, I + (F1 −

F0) + (F2 − F1) = I + (F2 − F0) is in the margin account. Likewise, on day 3,

I + (F1 − F0) + (F2 − F1) + (F3 − F2) = I + (F3 − F0) is in the margin account.

On day i, there is I +(Fi −F0) in the margin account. If this sum is smaller than

the maintenance margin then the investor gets a call and has to put the amount

of F0 − Fi into the margin account. Thus, till the maturity (day n), the margin

account payment will be F0 −Fn. As the futures price at maturity is the same as

the spot price on the day, Sn, the payment is F0−Sn which is exactly the futures

payment which is the difference between the futures price F0 on day 0 and the

spot price Sn at maturity.

2.5 Relation between forward price and futures price

We can prove that when the risk-free interest rate is constant, the forward price

is the same as the futures price for a contract when their delivery dates are same.


Proof

Assume that the delivery date of a futures and a forward contract is n days

away. Let Fi and Gi denote respectively the futures and forward prices at the end

of the ith day, 0 ≤ i ≤ n. The current forward and futures prices are respectively

Go and Fo and the forward price and futures price on maturity have to be the

same as the spot price, Sn, on the day; Fn = Gn = Sn.

Suppose an investor’s portfolio:

• Start a long position in eδ at the end of day 0

• Increase the long position to e2δ at the end of day 1

• Increase the long position to e3δ at the end of day 2

• · · ·.

When her(his) position of futures contracts is changed everyday, the value of the

entire investment at the end of day n is

n∑

i=1

ai(Fi − Fi−1)e(n−i)δ (2.17)

where

δ : Risk-free interest rate per day

ai : Number of futures contracts at the i − 1th day

ai(Fi − Fi−1) : Profit or loss from the position on day i

ai(Fi −Fi−1) earns the risk-free interest till the end of the futures life in the case

of profit. In the case of loss, the investor has to borrow ai|Fi − Fi−1| from a

bank and has to repay the loan and the interest. This is why we have e(n−i)δ in

Eq.(2.17).

For the investor’s portfolio, ai is eiδ so that Eq.(2.17) becomes

n∑

i=1

(Fi − Fi−1)enδ = (Fn − Fo)e

nδ. (2.18)


We know that the futures price at its maturity must be the spot price, Sn, for

the underlying asset. So the value of the investment (2.18) becomes

(Sn − Fo)enδ (2.19)

Let us consider an investor who has the above portfolio and the initial cash Fo

in the risk-free account. His(Her) money grows to Foenδ in n days. Thus his(her)

portfolio is worth

(Sn − Fo)enδ + Foe

nδ = Snenδ

on Day n.

Let us consider another investor who is in a long position of enδ underlying

assets. On Day n, the asset price is Sn so his(her) portfolio is worth Snenδ. We

realise that the two portfolios bear the same value on Day n thus they should

have the same value at any time. Therefore,

Soenδ = Fo.

The Left-hand side is the forward price. We have proved that the forward price

is the same as the futures price.

2.6 Hedging using futures

Let us consider an airline company which knows that it will buy 2 million gallons

of jet fuel in three months. The fuel price is not stable and the company does not

want to expose to the risk. However, unfortunately, there is no jet fuel futures

contract. The company tries to hedge the risk by the futures contracts on heating

oil. How many oil futures contracts would you recommend it to buy to hedge the

risk?

The company is short the fuel which is hedged by creating a long position in

oil futures. Buying h numbers of futures contracts, the change in the value of

company’s position is

δπ = −δS + hδF (2.20)


where δS is the change in spot price, S, and δF the change in futures price. For

example, suppose S = £203 at t0, S = £205 at t1, and S = £200 at t2 then

δS = £2 between t0 and t1 and δS = −£5 between t1 and t2.

Any price change happens in a random manner. For the random

price change of δS we define the mean as follows

δ̄S = 〈δS〉 =∑

i

piδSi

where pi is the probability of δSi. The mean value (or average

value) is sometimes called the expectation value and denoted by

E(δS). The standard deviation tells us the size of the fluctuation

in the price change:

σδS =

√

∑

i

pi(δSi − δ̄S)2 =

√

∑

i

piδS2i − (

∑

i

piδSi)2.

Another term very often used is the ‘variance’. The variance is the

square of the standard deviation. One more thing which appears

often in the study of statistics is ‘marginal’. Let us assume we

have more than one random variables. In this case, the marginal

mean value for variable 1, for example, is to consider the mean

value for variable 1 regardless of other variables.

We know that δS and δF are correlated random variables. By hedging we

want to minimise the variance of the company’s hedging position. To do it, the

company has to decide how many futures contracts it ought to buy. How?

• Step 1

Find the correlation between δS and δF .


• Step 2

Find the variance of the change in the value of the hedged position.

• Step 3

Find the minimum point of the variance with regard to the number of

futures contracts

To do the step 1, we have to know how the correlation between two random

variables is defined.

Characterisations of correlation between two random variables

1. Definition

Take a set of two random variables X = (X1,X2), the covariance between

the two variables is defined by

σ12(X) = E[(X1 − X̄1)(X2 − X̄2)]. (2.21)

2. Definition

For X = (X1,X2), its correlation coefficient is defined by

ρ(X) = E

[(

X1 − X̄1

σ1

)(

X2 − X̄2

σ2

)]

=σ12(X)

σ1σ2. (2.22)

where σ1 and σ2 are the standard deviations of X1 and X2. The correlation

coefficient has upper and lower bounds of 0 and 1: If the two are perfectly

correlated, ρ = 1 and if the two are totally independent, ρ = 0.

The variance σ2Y of a random variable Y = a1X1 + a2X2 is

σ2Y = a2

1σ2X1

+ a22σ

2X2

+ 2a1a2ρ(X1,X2)σX1σX2

(2.23)

where σ? is the standard deviation of the variable ?.

Assuming ρ the coefficient of correlation between δS and δF , we find the vari-

ance σ2hdg, of the change of the hedged position

σ2δπ = σ2

δS + h2σ2δF − 2hρσδSσδF . (2.24)


This is a quadratic equation with regard to h which is represented by a concave

parabola. Its derivative with regard to h is zero when

h = ρσδS

σδF(2.25)

which minimises σ2δπ. This makes a very good sense as h is larger when the two

prices are more correlated, i.e. ρ larger, the asset price fluctuates more, i.e. σS

larger, and the futures price fluctuates less, i.e. σF smaller.

example

An airliner will need 2,000,000 gallons of jet fuel in the future. The standard

deviations for the fluctuations of jet fuel price and heating oil price are respec-

tively σδS = 0.3 and σδF = 0.2 and their correlation coefficient is ρ = 0.8 then

h = 1.2. If a futures contract is on 1,000 gallons of heating oils we need to take

a long position on

1.2 ×2, 000, 000

1, 000= 2, 400

number of futures contracts.

Rolling the hedge forward

Let us consider the following example. In February, a company realised that

it would need 10,000 ounces of gold in July. The company decided to buy fu-

tures contracts on gold to hedge its risk. The company longs June contracts in

February. In May, it rolls the hedge forward into September contracts.

The price of gold is £180 in February and rises to £230 in July. The June

futures price rises from £190 in February to £210 in May. The September futures

price rises from £215 in May to £233 in July. With the hedging, how much cash

does the company need for the purchase of 10,000 ounces of gold?

2.7 Hedging using index futures

Let us assume that a portfolio is composed of shares of a few companies. To

hedge the portfolio we can either buy forward contracts of each share or buy


stock index futures. We can calculate how many stock index futures we have to

buy to optimise the hedging.

chap2

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