We ignore the margin requirement on the futures contract and treat the cash flow involved in establishing the futures

position as zero for the two reasons mentioned above: First, the margin is small relative to the amount of gold con-

trolled by one contract; and second, and more importantly, the margin requirement may be satisfied with interest-

bearing securities. For example, the investor merely needs to transfer Treasury bills already owned into the brokerage

account. There is no time-value-of-money cost.

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The initial cash flow of strategy A is negative, reflecting the cash outflow necessary to pur-

chase the gold at the current spot price, S0. At time T, the gold will be worth ST.

Strategy B involves an initial investment equal to the present value of the futures price that

will be paid at the maturity of the futures contract. By time T, the investment will grow to F0.

In addition, the profits to the long position at time T will be ST F0. The sum of the two com-

ponents of strategy B will be ST dollars, exactly enough to purchase the gold at time T regard-

less of its price at that time.

Each strategy results in an identical value of ST dollars at T. Therefore, the cost, or initial

cash outflow, required by these strategies also must be equal; it follows that

rf )T

F0 /(1 S0

or

rf )T

F0 S0(1 (16.1)

This gives us a relationship between the current price and the futures price of the gold. The

interest rate in this case may be viewed as the “cost of carrying” the gold from the present to

time T. The cost in this case represents the time-value-of-money opportunity cost”instead of

investing in the gold, you could have invested risklessly in Treasury bills to earn interest

income.

Suppose that gold currently sells for $280 an ounce. If the risk-free interest rate is 0.5% per

month, a six-month maturity futures contract should have a futures price of

EXAMPLE 16.6

rf)T $280(1.005)6

F0 S0(1 $288.51

Futures Pricing

If the contract has a 12-month maturity, the futures price should be

$280(1.005)12

F0 $297.27

If Equation 16.1 does not hold, investors can earn arbitrage profits. For example, suppose

the six-month maturity futures price in Example 16.6 were $289 rather than the “appropri-

ate” value of $288.51 that we just derived. An investor could realize arbitrage profits

by pursuing a strategy involving a long position in strategy A (buy the gold) and a short

position in strategy B (sell the futures contract and borrow enough to pay for the gold

purchase).

Initial Cash Flow at Time T

Action Cash Flow (6 months)

$280(1.005)6

Borrow $280, repay with interest at time T $280 $288.51

Buy gold for $280 280 ST

Enter short futures position (F0 $289) 0 289 ST

Total $0 $0.49

The net initial investment of this strategy is zero. Moreover, its cash flow at time T is positive

and riskless: The total payoff at time T will be $0.49 regardless of the price of gold. (The profit

is equal to the mispricing of the futures contract, $289 rather than $288.51.) Risk has been

eliminated because profits and losses on the futures and gold positions exactly offset each

other. The portfolio is perfectly hedged.

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582 Part FIVE Derivative Markets

Such a strategy produces an arbitrage profit”a riskless profit requiring no initial net in-

vestment. If such an opportunity existed, all market participants would rush to take advantage

of it. The results? The price of gold would be bid up, and/or the futures price offered down,

until Equation 16.1 is satisfied. A similar analysis applies to the possibility that F0 is less than

$288.51. In this case, you simply reverse the above strategy to earn riskless profits. We con-

clude, therefore, that in a well-functioning market in which arbitrage opportunities are com-

peted away, F0 S0(1 rf)T.

>

5. Return to the arbitrage strategy just laid out. What would be the three steps of the

Concept

strategy if F0 were too low, say $288? Work out the cash flows of the strategy now

CHECK and at time T in a table like the one on the previous page.

The arbitrage strategy can be represented more generally as follows:

Action Initial Cash Flow Cash Flow at Time T

rf)T

1. Borrow S0 S0 S0(1

2. Buy gold for S0 S0 ST

3. Enter short futures position 0 F0 “ ST

rf)T

Total 0 F0 S0(1

The initial cash flow is zero by construction: The money necessary to purchase the stock

in step 2 is borrowed in step 1, and the futures position in step 3, which is used to hedge the

value of the stock position, does not require an initial outlay. Moreover, the total cash flow to

the strategy at time T is riskless because it involves only terms that are already known when

the contract is entered. This situation could not persist, as all investors would try to cash in

on the arbitrage opportunity. Ultimately prices would change until the time T cash flow was

reduced to zero, at which point F0 would equal S0(1 rf)T. This result is called the spot-

spot-futures

futures parity theorem or cost-of-carry relationship; it gives the normal or theoretically

parity theorem,

correct relationship between spot and futures prices.

or cost-of-carry

We can easily extend the parity theorem to the case where the underlying asset provides a

relationship

flow of income to its owner. For example, consider a futures contract on a stock index such as

Describes the

the S&P 500. In this case, the underlying asset (i.e., the stock portfolio indexed to the S&P 500

theoretically correct

index), pays a dividend yield to the investor. If we denote the dividend yield as d, then the net

relationship between

spot and futures cost of carry is only rf d; the foregone interest earnings on the wealth tied up in the stock is

prices. Violation of offset by the flow of dividends from the stock. The net opportunity cost of holding the stock

the parity relationship

is the foregone interest less the dividends received. Therefore, in the dividend-paying case, the

gives rise to arbitrage

spot-futures parity relationship is6

opportunities.

d)T

F0 S0(1 rf (16.2)

where d is the dividend yield on the stock. Problem 8 at the end of the chapter leads you

through a derivation of this result.

The arbitrage strategy just described should convince you that these parity relationships are

more than just theoretical results. Any violations of the parity relationship give rise to arbi-

trage opportunities that can provide large profits to traders. We will see shortly that index ar-

bitrage in the stock market is a tool used to exploit violations of the parity relationship for

stock index futures contracts.

6

This relationship is only approximate in that it assumes the dividend is paid just before the maturity of the contract.

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Suppose that the risk-free interest rate is 0.5% per month, the dividend yield on the stock in-

dex is 0.1% per month, and the stock index is currently at 1,200. The net cost of carry is

EXAMPLE 16.7

therefore 0.5% 0.1% 0.4% per month. Given this, a three-month contract should have

a futures price of 1,200(1.004)3 1,214.46, while a six-month contract should have a fu- Stock Index

tures price of 1,200(1.004)6 1,229.09. If the index rises to 1,210, both futures prices will Futures Pricing

rise commensurately: The three-month futures price will rise to 1,210(1.004)3 1,224.58,

while the six-month futures price will rise to 1,210(1.004)6 1,239.33.

Spreads

Just as we can predict the relationship between spot and futures prices, there are similar ways

to determine the proper relationships among futures prices for contracts of different maturity

dates. Equation 16.2 shows that the futures price is in part determined by time to maturity. If

rf d, as is usually the case for stock index futures, then the futures price will be higher on

longer-maturity contracts. You can easily verify this by examining Figure 16.2, which includes

Wall Street Journal listings of several stock index futures contracts. (A warning to avoid con-

fusion. You will notice that for some stock index contracts in Figure 16.2, such as the S&P 500

contract, the financial pages omit decimal points to save space. Thus, the settlement price for

the March maturity contract should be interpreted as 1,149.00.) For futures on assets like gold,

which pay no “dividend yield,” we can set d 0 and conclude that F must increase as time to

maturity increases.

Equation 16.2 shows that futures prices should all move together. It is not surprising that

futures prices for different maturity dates move in unison, for all are linked to the same spot

price through the parity relationship. Figure 16.5 plots futures prices on gold for three matu-

rity dates. It is apparent that the prices move in virtual lockstep and that the more distant de-

livery dates require higher futures prices, as Equation 16.2 predicts.

F I G U R E 16.5

285

Gold futures prices,

October 2000

280

Futures prices, $/ounce

275

270

December 2000 delivery

265 February 2001 delivery

April 2001 delivery

260

10

11

12

13

16

17

18

19

20

23

24

25

26

27

30

31

2

3

4

5

6

9

Dates in October

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584 Part FIVE Derivative Markets

16.5 FINANCIAL FUTURES

Although futures markets have their origins in agricultural commodities, today™s market is

dominated by contracts on financial assets, as we saw in Figure 16.1.

Stock Index Futures

Currently, there are no stock futures on individual shares; futures trade instead on stock mar-

ket indexes such as the Standard & Poor™s 500. In contrast to most futures contracts, which

call for delivery of a specified commodity, these contracts are settled by a cash amount equal

to the value of the stock index in question on the contract maturity date times a multiplier that

scales the size of the contract. This cash settlement duplicates the profits that would arise with

actual delivery.

There are several stock index futures contracts currently traded. Table 16.2 lists some con-

tracts on major indexes, showing under contract size the multiplier used to calculate contract

settlements. An S&P 500 contract with an initial futures price of 1,200 and a final index value

TA B L E 16.2