
DETERMINATION OF FORWARDAND FUTURES PRICES
In this chapter we examine how forward prices and futures prices are related to the spot price ofthe underlying asset. Forward contracts are easier to analyze than futures contracts because thereis no daily settlement—only a single payment at maturity. Consequently, most of the analysis inthe first part of the chapter is directed toward determining forward prices rather than futuresprices. Luckily it can be shown that the forward price and futures price of an asset are usually veryclose when the maturities of the two contracts are the same. In the second part of the chapter weuse this result to examine the properties of futures prices for contracts on stock indices, foreignexchange, and other assets.
3.1 INVESTMENT ASSETS vs. CONSUMPTION ASSETS
 
 
When considering forward and futures contracts, it is important to distinguish hetween investmentassets and consumption assets. An investment asset is an asset that is held for investment purposesby significant numbers of investors. Stocks and bonds are clearly investment assets. Gold and silverare also examples of investment assets. Note that investment assets do not have to be heldexclusively for investment. Silver, for example, has a number of industrial uses. However, theydo have to satisfy the requirement that they are held by significant numbers of investors solely forinvestment. A consumption asset is an asset that is held primarily for consumption. It is not usuallyheld for investment. Examples of consumption assets are commodities such as copper, oil, andpork bellies.
As we will see later in this chapter, we can use arbitrage arguments to determine the forward andfutures prices of an investment asset from its spot price and other observable market variables. Wecannot do this for the forward and futures prices of consumption assets.
3.2 SHORT SELLING
 
 
Some of the arbitrage strategies presented in this chapter involve short selling. This trade, usuallysimply referred to as “shorting”, involves selling an asset that is not owned. It is something that ispossible for some—but not all—-investment assets. We will illustrate how it works by considering ashort sale of shares of a stock.
. A42 CHAPTER 3
Suppose an investor instructs a broker to short 500 IBM shares. The broker will carry out theinstructions by borrowing the shares from another client and selling them in the market in theusual way. The investor can maintain the short position for as long as desired, provided there arealways shares for the broker to borrow. At some stage, however, the investor will close out theposition by purchasing 500 IBM shares. These are then replaced in the account of the client fromwhom the shares were borrowed. The investor takes a profit if the stock price has declined and aloss if it has risen. If, at any time while the contract is open, the broker runs out of shares toborrow, the investor is short-squeezed and is forced to close out the position immediately even ifnoi ready to do so.
An investor with a short position must pay to the broker any income, such as dividends orinterest, that would normally be received on the securities that have been shorted. The broker willtransfer this to the account of the client from whom the securities have been borrowed. Considerthe position of an investor who shorts 500 IBM shares in April when the price per share is $120and closes out the position by buying them back in July when the price per share is $100. Supposethat a dividend of $1 per share is paid in May. The investor receives 500 x $120 = $60,000 in Aprilwhen the short position is initiated. The dividend leads to a payment by the investor of500 x $1 = $500 in May. The investor also pays 500 x $100 = $50,000 when the position is closedout in July. The net gain is therefore
$60,000 — $500 — $50,000 = $9,500
Regulators in the United States currently allow a stock to be shorted only on an uptick—that is,when the most recent movement in the price of the stock was an increase. An exception is madewhen traders are shorting a basket of stocks replicating a stock index.
3.3 MEASURING INTEREST RATES
 
 
 
Before getting into the details of how forward and futures prices are determined, it is important totalk about how interest rates are measured. A statement by a bank that the interest rate on one-yeardeposits is 10% per annum may sound straightforward and unambiguous. In fact, its precisemeaning depends on the way the interest rate is measured.If the interest rate is quoted with annual compounding, the bank’s statement that the interest rateis 10% means that $100 grows to$100 x 1.1 = $110
at the end of one year. When the interest rate is expressed with semiannual compounding, it meansthat we earn 5% every six months, with the interest being reinvested. In this case $100 grows to
$100 x 1.05 x 1.05 = $110.25
at the end of one year. When the interest rate is expressed with quarterly compounding, the bank’sstatement means that we earn 2.5% every three months, with the interest being reinvested. The$100 then grows to
$100 x 1.0254 = $110.38
at the end of one year. Table 3.1 shows the effect of increasing the compounding frequency further.The compounding frequency defines the units in which an interest rate is measured. A rateDetermination of Forward and Futures Prices 43
Table 3.1 Effect of the compounding frequency on thevalue of $100 at the end of one year when the interestrate is 10% per annum
 
Value of $100Compounding at end of yearfrequency (3)Annually (m = 1) 110.00Semiannually (7m = 2) 110.25Quarterly Ga = 4) 110.38Monthly (m = 12) 110.47Weekly (m = 52) 110.51Daily (m = 365) 110.52
expressed with one compounding frequency can be converted into an equivalent rate with adifferent compounding frequency. For example, from Table 3.1 we see that 10.25% with annualcompounding is equivalent to 10% with semiannual compounding. We can think of the differencebetween one compounding frequency and another to be analogous to the difference betweenkilometers and miles. They are two different units of measurement.
To generalize our results, suppose that an amount A is invested for n years at an interest rate ofR per annum. If the rate is compounded once per annum, the terminal value of the investment is
A(1 + R)"
If the rate is compounded m times per annum, the terminal value of the investment is{oRA (! + nm) (3.1)m
Continuous Compounding
The limit as m tends to infinity is known as continuous compounding. With continuous compound-ing, it can be shown that an amount A invested for n years at rate R grows to
Ae®™ (3.2)
where ¢ = 2.71828. The function e¢* is built into most calculators, so the computation of theexpression in equation (3.2) presents no problems. In the example in Table 3.1, A = 100, n = 1,and R = 0.1, so that the value to which A grows with continuous compounding is
1002"! = $110.52
This is (to two decimal places) the same as the value with daily compounding. For most practicalpurposes, continuous compounding can be thought of as being equivalent to daily compounding.Compounding a sum of money at a continuously compounded rate R for n years involvesmultiplying it by e®". Discounting it at a continuously compounded rate R for n years involvesmultiplying by e7*".
In this book interest rates will be measured with continuous compounding except where44 CHAPTER 3
otherwise stated. Readers used to working with interest rates that are measured with annual,semiannual, or some other compounding frequency may find this a little strange at first. However,continuously compounded interest rates are used to such a great extent in pricing derivatives that itmakes sense to get used to working with them now.
Suppose that R, is a rate of interest with continuous compounding and R,, is the equivalent ratewith compounding m times per annum. From the results in equations (3.1) and (3.2), we must have
R mitRon AmAe =a(t+ 2)
ek = ( + an)m
Ry,R.= min( +4) (3.3)
orThis means that
andRy = m(eRe’™ — 4) (3.4)
These equations can be used to convert a rate with a compounding frequency of m times perannum to a continuously compounded rate and vice versa. The function In is the natural logarithmfunction and is built into most calculatrs. It is defined so that if y = Inx, then x =e’.
Example 3.1 Consider an interest rate that is quoted as 10% per annum with semiannualcompounding. From equation (3.3) with m = 2 and R,, = 0.1, the equivalent rate with continuouscompounding is
or 9.758% per annum.
Example 3.2 Suppose that a lender quotes the interest rate on loans as 8% per annum withcontinuous compounding, and that interest is actually paid quarterly. From equation (3.4) withm=4and R, = 0.08, the equivalent rate with quarterly compounding is
4(e5°8/4 _ 1) = 0.0808
or 8.08% per annum. This means that on a $1,000 loan, interest payments of $20.20 would berequired each quarter,
3.4 ASSUMPTIONS AND NOTATION
 
 
 
 
In this chapter we will assume that the following are all true for some market participants:
1. The market participants are subject to no transactions costs when they trade.2. The market participants are subject to the same tax rate on all net trading profits.
3. The market participants can borrow money at the same risk-free rate of interest as they canlend money.
4. The market participants take advantage of arbitrage opportunities as they occur.Determination of Forward and Futures Prices 45
Note that we do not require these assumptions to be true for all market participants. All that werequire is that they be true—or at least approximately true—for a few key market participants suchas large investment banks. This is not unreasonable. It is the trading activities of these key marketparticipants and their eagerness to take advantage of arbitrage opportunities as they occur thatdetermine the relationship between forward and spot prices.
The following notation will be used throughout this chapter:
e fohowimeg notation be used throughont this cn:
T: Time until delivery date in a forward or futures contract (in years)So: Price of the asset underlying the forward or futures contract todayFy: Forward or futures price today
r: Risk-free rate of interest per annum, expressed with continuous compounding, for aninvestment maturing at the delivery date (i.e., in 7 years)
The risk-free rate, r, is in theory the rate at which money is borrowed or lent when there is no creditrisk, so that the money is certain to be repaid. It is often thought of as the Treasury rate, that is, therate at which a national government borrows in its own currency. In practice, large financialinstitutions usually set r equal to the London Interbank Offer Rate (LIBOR) instead of Treasuryrate in the formulas in this chapter and in those in the rest of the book. LIBOR will be discussed inChapter 5.
3.6 FORWARD PRICE FOR AN INVESTMENT ASSET
 
 
The easiest forward contract to value is one written on an investment asset that provides the holderwith no income. Non-dividend-paying stocks and zero-coupon bonds are examples of suchinvestment assets.!
Illustration
Consider a long forward contract to purchase a non-dividend-paying stock in three months.Assume the current stock price is $40 and the three-month risk-free interest rate is 5% per annum.We consider strategies open to an arbitrageur in two extreme situations.
Suppose first that the forward price is relatively high at $43. An arbitrageur can borrow $40 atthe risk-free interest rate of 5% per annum, buy one share, and short a forward contract to sell oneshare in three months. At the end of the three months, the arbitrageur delivers the share andreceives $43. The sum of money required to pay off the joan is
40¢9°*3/12 — $40.50
By following this strategy, the arbitrageur locks in a profit of $43.00 — $40.50 = $2.50 at the end ofthe three-month period.
Suppose next that the forward price is relatively low at $39. An arbitrageur can short one share,invest the proceeds of the short sale at 5% per annum for three months, and take a long position in
' Some of the contracts mentioned in the first half of this chapter (e.g., forward contracts on individual stocks) donot normally arise in practice. However, they form useful examples for developing our ideas.46 CHAPTER 3
 
a three-month forward’contract. The proceeds of the short sale grow to
409 89*3/12
or $40.50 in three months. At the end of the three months, the arbitrageur pays $39, takes deliveryof 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 therefore made at the end of the three months.
Under what circumstances do arbitrage opportunities such as those we have just considered exist.The first arbitrage works when the forward price is greater than $40.50. The second arbitrageworks when the forward price is less than $40.50. We deduce that for there to be no arbitrage theforward price must be exactly $40.50.
A Generalization
To generalize this example, we consider a forward contract on an investment asset with price Spthat provides no income. Using our notation, T is the time to maturity, r is the risk-free rate, andFy is the forward price. The relationship between Fy and So is
Fy = Spe’? (3.5)
If Fy > Spe’, arbitrageurs can buy the asset and short forward contracts on the asset. If Fy < Soe’?,they can short the asset and buy forward contracts on it.? In our example, Sy = 40, r = 0.05,and T = 0.25, so that equation (3.5) gives
Fy = 40¢%5*925 — $40.50which is in agreement with our earlier calculations.
Example 3.3 Consider a four-month forward contract to buy a zero-coupon bond that will matureone year from today. The current price of the bond is $930. (This means that the bond will haveeight months to go when the forward contract matures.) We assume that the four-month risk-freerate of interest (continuously compounded) is 6% per annum. Because zero-coupon bonds provideno income, we can use equation (3.5) with T = 4/12, r = 0.06, and Sy = 930. The forward price, Fo,is given by
Fy = 930e%*4/1? — $948.79
This would be the delivery price in a contract negotiated today.
What If Short Sales Are Not Possible?
Short sales are not possible for all investment assets. As it happens, this does not matter. To deriveequation (3.5), we do not need to be able to short the asset. All that we require is that there be asignificant number of people who hold the asset purely for investment (and by definition this is
> For another way of seeing that equation (3.5) is correct, consider the following strategy: buy one unit of the assetand enter into a short forward contract to sell it for Fy at time 7. This costs Sg and is certain to lead to a cash inflowof Fy at time 7. Therefore Sy must equal the present value of Fp; that is, Sp = Foe", or equivalently Fy = Soe’”.Determination of Forward and Futures Prices 47
always true of an investment asset). If the forward price is too low, they will find it attractive to sellthe asset and take a long position in a forward contract.
Suppose the underlying asset is gold and assume no storage costs or income.* If Fy > Spe””, aninvestor can adopt the following strategy:
1. Borrow Sp dollars at an interest rate r for T years.
2. Buy one ounce of gold.
3. Short a forward contract on one ounce of gold.At time T one ounce of gold is sold for Fy. An amount Sye’’ is required to repay the loan at thistime and the investor makes a profit of Fy — Spe’.
Suppose next that Fy < Spe’. In this case an investor who owns one ounce of gold can
1. Sell the gold for Sp.
2. Invest the proceeds at interest rate r for time T.
3. Take a long position in a forward contract on one ounce of gold.At time 7 the cash invested has grown to Sye’’. The gold is repurchased for Fy and the investormakes a profit of Spe’ — Fo retative iv the position the invesior would have been in if the gold hadbeen kept.
As in the non-dividend-paying stock example considered earlier, we can expect the forward price
to adjust so that neither of the two arbitrage opportunities we have considered exists. This meansthat the relationship in equation (3.5) must hold.
 
 
 
In this section we consider a forward contract on an investment asset that will provide a perfectlypredictable cash income to the holder. Examples are stocks paying known dividends and coupon-bearing bonds. We adopt the same approach as in the previous section. We first look at anumerical example and then review the formal arguments.
Illustration
Consider a long forward contract to purchase a coupon-bearing bond whose current price is $900.We will suppose that the forward contract matures in one year and the bond matures in five years,so that the forward contract is a contract to purchase a four-year bond in one year. We will alsosuppose that coupon payments of $40 are expected after 6 months and 12 months, with the secondcoupon payment being immediately prior to the delivery date in the forward contract. We assumethe six-month and one-year risk-free interest rates (continuously compounded) are 9% per annumand 10% per annum, respectively.
Suppose first that the forward price is relatively high at $930. An arbitrageur can borrow $900 tobuy the bond and short a forward contract. The first coupon payment has a present value of40e~°-°*0'5 — $38.24. Of the $900, $38.24 is therefore borrowed at 9% per annum for six monthsso that it can be repaid with the first coupon payment. The remaining $861.76 is borrowed at 10%per annum for one year. The amount owing at the end of the year is. 861.76e"!™! = $952.39. The
} Section 1.3 provides a numerical example for this case.43 CHAPTER 3
second coupon provides $40 toward this amount, and $930 is received for the bond under theterms of the forward contract. The arbitrageur therefore makes a net profit of
$40 + $930 — $952.39 = $17.61
Suppose next that the forward price is relatively low at $905. An investor who holds the bond cansell it and enter into a long forward contract. Of the $900 realized from selling the bond, $38.24 isinvested for 6 months at 9% per annum so that it grows into an amount sufficient to equal thecoupon that would have been paid on the bond. The remaining $861.76 is invested for 12 monthsat 10% per annum and grows to $952.39. Of this sum, $40 is used to replace the coupon that wouldhave been received on the bond, and $905 is paid under the terms of the forward contract toreplace the bond in the investor’s portfolio. The investor therefore gains
$952.39 — $40.00 — $905.00 = $7.39
relative to the situation the investor would have been in by keeping the bond.The first strategy produces a profit when the forward price is greater than $912.39, whereas thesecond strategy produces a profit when the forward price is less than $912.39. It follows that if
O19 25
ihere are no arbitrage opportunities then ihe forward price musi be $912.39.
A Generalization
We can generalize from this example to argue that when an investment asset will provide incomewith a present value of / during the life of a forward contract
Fy = (Sp — De™ (3.6)In our example, Sy = 900.00, J = 40e °°*5 4 40e°!*! — 74,433, r= 0.1, and T = 1, so thatFy = (900.00 — 74.433)e"'*! = $912.39
This is in agreement with our earlier calculation. Equation (3.6) applies to any asset that provides aknown cash income.
If Fy > (Sp — De’, an arbitrageur can lock in a profit by buying the asset and shorting aforward contract on the asset. If Fy < (Sy — De” an arbitrageur can lock in a profit by shorting theasset and taking a long position in a forward contract. If short sales are not possible, investors whoown the asset will find it profitable to sell the asset and enter into long forward contracts.*
Example 3.4 Consider a 10-month forward contract on a stock with a price of $50. We assumethat the risk-free rate of interest (continuously compounded) is 8% per annum for all maturities. Wealso assume that dividends of $0.75 per share are expected after three months, six months, and ninemonths, The present value of the dividends, /, is given by
l= 0.757 0:08 3/12 + 0.75¢e7 0086/12 4 0.75 0089/12 = 2.162
4 For another way of seeing that equation (3.6) is correct, consider the following strategy: buy one unit of the assetand enter into a short forward contract to sell it for Fy at time T. This costs Sp and is certain to lead to a cash inflowof Fy at time 7 and income with a present value of /. The initial outflow is Sy. The present value of the inflows isI+ Foe”, Hence, Sy =/+ Foe”, or equivalently Fy = (Sp — De’.Determination of Forward and Futures Prices 49
The variable 7 is 10 months, so that the forward price, Fo, from equation (3.6), is given byFy = (50 — 2.162)0%* 1? — $51.14If the forward price were less than this, an arbitrageur would short the stock spot and buy forward
contracts. If the forward price were greater than this, an arbitrageur would short forward contractsand buy the stock spot.
3.7 KNOWN YIELD
 
 
We now consider the situation where the asset underlying a forward contract provides a knownyield rather than a known cash income. This means that the income is known when expressed as apercent of the asset’s price at the time the income is paid. Suppose that an asset is expected toprovide a yield of 5% per annum. This could mean that income equal to 5% of the asset price ispaid once a year. (The yield would then be 5% with annual compounding.) It could mean thatincome equal to 2.5% of the asset price is paid twice a year. (The yield would then be 5% perannum with semiannual compounding.) In Section 3.3 we explained that we will normally measureinterest rates with continuous compounding. Similarly we will normally measure yields withcontinuous compounding. Formulas for translating a yield measured with one compoundingfrequency to a yield measured with another compounding frequency are the same as those givenfor interest rates in Section 3.3.
Define g as the average yield per annum on an asset during the life of a forward contract. It canbe shown (see Problem 3.22) that
Fy = Spe" (3.7)
Example 3.5 Consider a six-month forward contract on an asset that is expected to provide incomeequal to 2% of the asset price once during a six-month period. The risk-free rate of interest (withcontinuous compounding) is 10% per annum. The asset price is $25. In this case Sp = 25, r = 0.10,and T = 0.5. The yield is 4% per annum with semiannual compounding. From equation (3.3) this is3.96% per annum with continuous compounding. It follows that ¢ = 0.0396, so that from equation(3.7) the forward price Fp is given by
Fy — 25 ¢(0-10-0.0396)x0.5 = $25.77
3.8 VALUING FORWARD CONTRACTS
 
 
The value of a forward contract at the time it is first entered into is zero. At a later stage it mayprove to have a positive or negative value. Using the notation introduced earlier, we suppose Fy isthe current forward price for contract that was negotiated some time ago, the delivery date is in Tyears, and r is the 7-year risk-free interest rate. We also define:
K: Delivery price in the contractf: Value of a long forward contract today50 CHAPTER 3
A general result, applicable to all forward contracts (both those on investment assets and those onconsumption assets), isf = (Fy ~ Kye (3.8)
When the forward contract is first negotiated, K is set equal to Fy and f = 0. As time passes, boththe forward price, Fy, and the value of the forward contract, f, change.
To see why equation (3.8) is correct, we compare a long forward contract that has a deliveryprice of &) with an otherwise identical long forward contract that has a delivery price of K. Thedifference between the two is only in the amount that will be paid for the underlying asset at timeT. Under the first contract this amount is Fy; under the second contract it is K. A cash outflowdifference of Fy ~ K at time 7 translates to a difference of (Fy — K)e~”’ today. The contract with adelivery price Fy is therefore less valuable than the contract with delivery price K by an amount(Fy — K)e~. The value of the contract that has a delivery price of Fy is by definition zero. Itfollows that the value of the contract with a delivery price of K is (Fy — K)e’’. This provesequation (3.8). Similarly, the value of a short forward contract with delivery price K is
(K = Foe"
Example 3.6 A long forward contract on a non-dividend-paying stock was entered into some timeago. It currently has six months to maturity. The risk-free rate of interest (with continuous’compounding) is 10% per annum, the stock price is $25, and the delivery price is $24. In this caseSo = 25, r=0.10, 7 =0.5, and K = 24. From equation (3.5) the six-month forward price, Fo, isgiven by
Fy = 25e°!*° = $26.28
f = (26.28 — 24) = $2.17
Equation (3.8) shows that we can value a long forward contract on an asset by making theassumption that the price of the asset at the maturity of the forward contract equals the forwardprice Fy. To see this, note that when we make the assumption, a long forward contract provides apayoff at time T of Fy ~ K. This has a present value of (Fy — K)e~"?, which is the value of f inequation (3.8). Similarly, we can value a short forward contract on the asset by assuming that thecurrent forward price of the asset is realized.
Using equation (3.8) in conjunction with (3.5) gives the following expression for the value of aforward contract on an investment asset that provides no income:
f=S—Ke™ 3.9)
Similarly, using equation (3.8) in conjunction with (3.6) gives the following expression for the valueof a long forward contract on an investment asset that provides a known income with presentvalue [:
f=S)—-I-Ke™ (3.10)
Finally, using equation (3.8) in conjunction with (3.7) gives the following expression for the valueof a long forward contract on an investment asset that provides a known yield at rate ¢:
f = Spe! — Ke’ B.1DDetermination of Forward and Futures Prices 51
3.9 ARE FORWARD PRICES AND FUTURES PRICES EQUAL?
 
Appendix 3A provides an arbitrage argument to show that when the risk-free interest rate isconstant and the same for all maturities, the forward price for a contract with a certain deliverydate is the same as the futures price for a contract with that delivery date. The argument inAppendix 3A can be extended to cover situations where the interest rate is a known function oftime.
When interest rates vary unpredictably (as they do in the real world), forward and futures pricesare in theory no longer the same. We can get a sense of the nature of the relationship between thetwo by considering the situation where the price of the underlying asset, S, is strongly positivelycorrelated with interest rates. When S increases, an investor who holds a long futures positionmakes an immediate gain because of the daily settlement procedure. The positive correlationindicates that it is likely that interest rates have also increased. The gain will therefore tend to beinvested at a higher than average rate of interest. Similarly, when S decreases, the investor will incuran immediate loss. This loss will tend to be financed at a lower than average rate of interest. Aninvestor holding a forward contract rather than a futures contract is not affected in this way byinterest rate movements. It follows that a long futures contract will be more attractive than asimilar long forward contract. Hence, when S is strongly positively correlated with interest rates,futures prices will tend to be higher than forward prices. When S is strongly negatively correlatedwith interest rates, a similar argument shows that forward prices will tend to be higher than futuresprices.
The theoretical differences een forward andmonths are in most circumstances sufficiently small to be ignored. In practice, there are a numberof factors not reflected in theoretical models that may cause forward and futures prices to bedifferent. These include taxes, transactions costs, and the treatment of margins. The risk that thecounterparty will default is generally less in the case of a futures contract because of the role of theexchange clearinghouse. Also, in some instatices, futures contracts are more liquid and easier totrade than forward contracts. Despite all these points, for most purposes it is reasonable to assumethat forward and futures prices are the same. This is the assumption we will usually make in thisbook. We will use the symbol Fy to represent both the futures price and the forward price of anasset today.
As the life of a futures contract increases, the differences between forward and futures contractsare liable to become significant. It is then dangerous to assume that forward and futures prices areperfect substitutes for each other. This point is particularly relevant to Eurodollar futures contractsbecause they have maturities as long as ten years. These contracts are covered in Chapter 5.
ences between forward and futures prices or contracts that last only a few
ices Tor {rac nat only
Empirical Research
Some empirical research that has been carried out comparing forward and futures contracts islisted at the end of this chapter. Cornell and Reingan