s
Consider the position of the corporation in the trade we have just described. What are the possibleoutcomes? The forward contract obligates the corporation to buy £1 million for $1,435,900. If thespot exchange rate rose to, say, 1.5000, at the end of the six months the forward contract would beworth $64,100 (= $1,500,000 — $1,435,900) to the corporation. It would enable £1 million to bepurchased ai 1.4359 rather than 1.5060. Similarly, if ihe spot exchange rate fell to 1.4000 at the end ofthe six months, the forward contract would have a negative value to the corporation of $35,900because it would lead to the corporation paying $35,900 more than the market price for the sterling.In general, the payoff from a long position in a forward contract on one unit of an asset is
Sp—K
where K is the delivery price and S; is the spot price of the asset at maturity of the contract. This isbecause the holder of the contract is obligated to buy an asset worth S; for K. Similarly, the payofffrom a short position in a forward contract on one unit of an asset is
K—-Sy4 CHAPTER 1
 
4 Payoff 4 Payoff
 
(a) (b)
Figure1.1 Payoffs from forward contracts: (a) long position, (b) short position.Delivery price = K; price of asset at maturity = S,
These payoffs can be positive or negative. They are illustrated in Figure 1.1. Because it costsnothing to enter into a forward contract, the payoff from the contract is also the trader’s total gainor loss from the contract.
Forward Price and Delivery Price
It is important to distinguish between the forward price and delivery price. The forward price is themarket price that would be agreed to today for delivery of the asset at a specified maturity date.The forward price is usually different from the spot price and varies with the maturity date(see Table 1.1).
In the example we considered earlier, the forward price on August 16, 2001, is 1.4359 for acontract maturing on February 16, 2002. The corporation enters into a contract and 1.4359hecomes the delivery price for the contract. As we move through time the delivery price for thecorporation’s contract does not change, but the forward price for a contract maturing on February16, 2002, is likely to do so. For example, if GBP strengthens relative to USD in the second half ofAugust the forward price could rise to 1.4500 by September 1, 2001,
Forward Prices and Spot Prices
We will be discussing in some detail the relationship between spot and forward prices in Chapter 3.In this section we illustrate the reason why the two are related by considering forward contracts ongold. We assume that there are no storage costs associated with gold and that gold earns no income.!Suppose that the spot price of gold is $300 per ounce and the risk-free interest rate forinvestments lasting one year is 5% per annum. What is a reasonable value for the one-yearforward price of gold?! This is not totally realistic. In practice, storage costs are close to zero, but an income of | to 2% per annum can beearned by lending gold.Introduction 5
Suppose first that the one-year forward price is $340 per ounce. A trader can immediately takethe following actions:
1. Borrow $300 at 5% for one year.2. Buy one ounce of gold.3. Enter into a short forward contract to sell the gold for $340 in one year.
The interest on the $300 that is borrowed (assuming annual compounding) is $15. The trader can,therefore, use $315 of the $340 that is obtained for the goid in one year to repay the loan. Theremaining $25 is profit. Any one-year forward price greater than $315 will lead to this arbitragetrading strategy being profitable.
Suppose next that the forward price is $300. An investor who has a portfolio that includes gold can
1. Sell the gold for $300 per ounce.2. Invest the proceeds at 5%.3. Enter into a long forward contract to repurchase the gold in one year for $300 per ounce.
When this strategy is compared with the alternative strategy of keeping the gold in the portfolio forone year, we sce that the investor is better off by $15 per ounce. In any situation where the forwardprice is less than $315, investors holding gold have an incentive to sell the gold and enter into along forward contract in the way that has been described.
The first strategy is profitable when the one-year forward price of gold is greater than $315. Asmore traders attempt to take advantage of this strategy, the demand for short forward contractswill increase and the one-year forward price of gold will fall. The second strategy is profitable forall investors who hold gold in their portfolios when the one-year forward price of gold is less than$315. As these investors attempt to take advantage of this strategy, the demand for long forwardcontracts will increase and the one-year forward price of gold will rise. Assuming that individualsare always willing to take advantage of arbitrage opportunities when they arise, we can concludethat the activities of traders should cause the one-year forward price of gold to be exactly $315.Any other price leads to an arbitrage opportunity.”
“aA riivrit = airi ruil i
 
Like a forward contract, a futurcs contract is an agrecmont between two partics to buy or sell anasset at a certain time in the future for a certain price. Unlike forward contracts, futures contractsare normally traded on an exchange. To make trading possible, the exchange specifies certainstandardized features of the contract. As the two parties to the contract do not necessarily knoweach other, the exchange also provides a mechanism that gives the two parties a guarantee that thecontract will be honored.
The largest exchanges on which futures contracts are traded are the Chicago Board of Trade(CBOT) and the Chicago Mercantile Exchange (CME). On these and other exchanges throughoutthe world, a very wide range of commodities and financial assets form the underlying assets in thevarious contracts. The commodities include pork bellies, live cattle, sugar, wool, lumber, copper,aluminum, gold, and tin. The financial assets include stock indices, currencies, and Treasury bonds.
? Our arguments make the simplifying assumption that the rate of interest on borrowed funds is the same as the rateof interest on invested funds.6 CHAPTER 1
One way in which a futures contract is different from a forward contract is that an exact deliverydate is usually not specified. The contract is referred to by its delivery month, and the exchangespecifies the period during the month when delivery must be made. For commodities, the deliveryperiod is often the entire month. The holder of the short position has the right to choose the timeduring the delivery period when it will make delivery. Usually, conuracts with several differentdelivery months are traded at any one time. The exchange specifies the amount of the asset to bedelivered for one contract and how the futures price is to he quoted. In the case of a commadity,the exchange also specifies the product quality and the delivery location. Consider, for example, thewheat futures contract currently traded on the Chicago Board of Trade. The size of the contract is5,000 bushels. Contracts for five delivery months (March, May, July, September, and December)are available for up to 18 months into the future. The exchange specifies the grades of wheat thatcan be delivered and the places where delivery can be made.
Futures prices are regularly reported in the financial press. Suppose that on September |, theDecember futures price of gold is quoted as $300. This is the price, exclusive of commissions, atwhich traders can agree to buy or sell gold for December delivery. It is determined on the floor of theexchange in the same way as other prices (i.c., by the laws of supply and demand). If more traderswant to go long than to go short, the price goes up; if the reverse is true, the price goes down.
Further details on issues such as margin requirements, daily settlement procedures, deliveryprocedures, bid-offer spreads, and the role of the exchange clearinghouse are given in Chapter 2.
 
1.5 OPTIONS
Options are traded both on exchanges and in the over-the-counter market. There are two basictypes of options. A call option gives the holder the right to buy the underlying asset by a certain datefor a certain price. A put option gives the holder the right to sell the underlying asset by a certaindate for a certain price. The price in the contract is known as the exercise price or strike price; thedate in the contract is known as the expiration date or maturity. American options can be exercised atany time up to the expiration date. European options can be exercised only on the expiration dateitself. Most of the options that are traded on exchanges are American. In the exchange-tradedequity options market, one contract is usually an agreement to buy or sell 100 shares. Europeanoptions are generally easier to analyze than American options, and some of the properties of anAmerican option are frequently deduced from those of its European counterpart.
It should be emphasized that an option gives the holder the right to do something. The holderdoes not have to exercise this right. This is what distinguishes options from forwards and futures,where the holder is obligated to buy or sell the underlying asset. Note that whereas it costs nothing
to enter into a forward or futures contract, there is a cast to acquiring an option.
Call Options
Consider the situation of an investor who buys a European call option with a strike price of $60 topurchase 100 Microsoft shares. Suppose that the current stock price is $58, the expiration date of
3 In Chapter 3 we discuss the relationship between a futures price and the spot price of the underlying asset (gold, inthis case).
4 Note that the terms American and European do not refer to the location of the option or the exchange. Someoptions trading on North American exchanges are European.Introduction 7
 
 
Profit ($)3020 410Terminalstock price ($)0 + =90-5
 
Figure 1.2. Profit from buying a European call option on one Microsoft share.Option price = $5; strike price = $60
the option is in four months, and the price of an option to purchase one share is $5. The initialinvestment is $500. Because the option is European, the investor can exercise only on the expirationdate. If the stock price on this date is less than $60, the investor will clearly choose not to exercise.(There is no point in buying, for $60, a share that has a market value of less than $60.) In thesecircumstances, the investor loses the whole of the initial investment of $500. If the stock price isabove $60 on the expiration date, the option will be exercised. Suppose, for example, that the stockprice is $75. By exercising the option, the investor is able to buy 100 shares for $60 per share. If theshares are sold immediately, the investor makes a gain of $15 per share, or $1,500, ignoringtransactions costs. When the initial cost of the option is taken into account, the net profit to theinvestor is $1,000.
Figure 1.2 shows how the investor’s net profit or loss on an option to purchase one share varieswith the final stock price in the example. (We ignore the time value of money in calculating theprofit.) It is important to realize that an investor sometimes exercises an option and makes a lossoverall. Suppose that in the example Microsoft’s stock price is $62 at the expiration of the option.The investor would exercise the option for a gain of 100 x ($62 — $60) = $200 and realize a lossoverall of $300 when the initial cost of the option is taken into account. It is tempting to argue thatthe investor should not exercise the option in these circumstances. However, not exercising wouldlead to an overall loss of $500, which is worse than the $300 loss when the investor exercises. Ingeneral, call options should always be exercised at the expiration date if the stock price is above thestrike price.
Put Options
Whereas the purchaser of a call option is hoping that the stock price will increase, the purchaser of aput option is hoping that it will decrease. Consider an investor who buys a European put option tosell 100 shares in IBM with a strike price of $90. Suppose that the current stock price is $85, theexpiration date of the option is in three months, and the price of an option to sell one share is $7. Theinitial investment is $700. Because the option is European, it will be exercised only if the stock priceis below $90 at the expiration date. Suppose that the stock price is $75 on this date. The investor can8 CHAPTER 1
 
A Profit ($)
       
 
    
30 -20 -10FTerminalstock price ($)0 K 1 le _L 1 i i >60 70 100 110 120
—7TF
 
Figure 1.3 Profit from buying a European put option on one IBM share.Option price = $7; strike price = $90
buy 100 shares for $75 per share and, under the terms of the put option, sell the same shares for $90
to realize a gain of $15 per share, or $1,500 (again transactions costs are ignored). When the $700
initial cost of the option is taken into account, the investor’s net profit is $800. There is no guaranteethat the investor will make a gain. If the final stock price is above $90, the put option expiresworthless, and the investor loses $700. Figure 1.3 shows the way in which the investor’s profit or losson an option to sell one share varies with the terminal stock price in this example.
Early Exercise
As already mentioned, exchange-traded stock options are usually American rather than European.That is, the investor in the foregoing examples would not have to wait until the expiration date beforeexercising the option. We will see in later chapters that there are some circumstances under which it isoptimal to exercise American options prior to maturity.
Option Positions
There are two sides to every option contract. On one side is the investor who has taken the longposition (i.e., has bought the option). On the other side is the investor who has taken a shortposition (i.c., has sold or written the option). The writer of an option receives cash up front, buthas potential liabilities later. The writer’s profit or loss is the reverse of that for the purchaser of theoption. Figures 1.4 and 1.5 show the variation of the profit or loss with the final stock price forwriters of the options considered in Figures 1.2 and 1.3.
There are four types of option positions:
1. A long position in a call option.2. A long position in a put option.3. A short position in a call option.4. A short position in a put option.Introduction 9
       
 
Profit ($)570 80 900 1 Lo 1 >30 40 50 60 Terminal
stock price ($)-10-20-30
Figure 1.4 Profit from writing a European call option on one Microsoft share.Option price = $5; strike price = $60
It is often useful to characterize European option positions in terms of the terminal! value or payoffto the investor at maturity. The initial cost of the option is then not included in the calculation. IfK is the strike price and S; is the final price of the underlying asset, the payoff from a long positionin a European call option is
maa(Sp — K, 0)
This reflects the fact that the option will be exercised if S; > K and will not be exercised if Sp < K.The payoff to the holder of a short position in the European call option is
~—max(S7 — K, 0) = min(K — Sp, 0)
A Profit ($)
Terminalstock price ($)
 
 
Figure 1.5 Profit from writing a European put option on one IBM share.Option price = $7; strike price = $9010 CHAPTER 1
 
 
A Payoff 4 PayoffSp SpK K(a) (b)4 Payoff A PayoffSr SpK J K(c) (d)
Figure 1.6 Payoffs from positions in European options: (a) long call, (b) short call, (c) long put,(d) short put. Strike price = K; price of asset at maturity = S;
The payoff to the holder of a long position in a European put option ismax(K — Sr, 0)and the payoff from a short position in a European put option is
—max(K — Sp, 0) = min(S; — K, 0)Figure 1.6 shows these payoffs.
1.6 TYPES OF TRADERS
 
 
Derivatives markets have been outstandingly successful. The main reason is that they haveattracted many different types of traders and have a great deal of liquidity. When an investorwants to take one side of a contract, there is usually no problem in finding someone that isprepared to take the other side.
Three broad categories of traders can be identified: hedgers, speculators, and arbitrageurs.Hedgers use futures, forwards, and options to reduce the risk that they face from potential futuremovements in a market variable. Speculators use them to bet on the future direction of a marketvariable. Arbitrageurs take offsetting positions in two or more instruments to lock in a profit. Inthe next few sections, we consider the activities of each type of trader in more detail.Introduction 11
Hedgers
We now illustrate how hedgers can reduce their risks with forward contracts and options.
Suppose that it is August 16, 2001, and ImportCo, a company based in the United States, knowsthat it will pay £10 million on November 16, 2001, for goods it has purchased from a British supplier.The USD-GBP exchange rate quotes made by a financial institution are given in Table 1.1.ImportCo could hedge its foreign exchange risk by buying pounds (GBP) from the financialinstitution in the three-month forward market at 1.4407. This would have the effect of fixing theprice to be paid to the British exporter at $14,407,000.
Consider next another U.S. company, which we will refer to as ExportCo, that is exportinggoods to the United Kingdom and on August 16, 2001, knows that it will receive £30 million threemonths later. ExportCo can hedge its foreign exchange risk by selling £30 million in the three-month forward market at an exchange rate of 1.4402. This would have the effect of locking in theUS. dollars to be realized for the sterling at $43,206,000.
Note that if the companies choose not to hedge they might do better than if they do hedge.Alternatively, they might do worse. Consider ImportCo. If the exchange rate is 1.4000 on November16 and the company has not hedged, the £10 million that it has to pay will cost $14,000,000, which isless than $14,407,000. On the other hand, if the exchange rate is 1.5000, the £10 million will cost$15,000,000—and the company will wish it had hedged! The position of ExportCo if it does nothedge is the reverse. If the exchange rate in September proves to be less than 1.4402, the company willwish it had hedged; if the rate is greater than 1.4402, it will be pleased it had not doue so.
This example illustrates a key aspect of hedging. The cost of, or price received for, the underlyingasset is ensured. However, there is no assurance that the outcome with hedging will be better thanthe outcome without hedging.
Options can also be used for hedging. Consider an investor who in May 2000 owns 1,000Microsoft shares. The current share price is $73 per share. The investor is concerned that thedevelopments in Microsoft’s antitrust case may cause the share price to decline sharply in the nexttwo months and wants protection. The investor could buy 10 July put option contracts with a strikeprice of $65 on the Chicago Board Options Exchange. This would give the investor the right to sell1,000 shares for $65 per share. If the quoted option price is $2.50, each option contract would cost100 x $2.50 = $250, and the total cost of the hedging strategy would be 10 x $250 = $2,500.
The strategy costs $2,500 but guarantees that the shares can be sold for at least $65 per shareduring the life of the option. If the market price of Microsoft falls below $65, the options can beexercised so that $65,000 is realized for the entire holding. When the cost of the options is takeninto account, the amount realized is $62,500. If the market price stays above $65, the options arenot exercised and expire worthless. However, in this case the value of the holding is always above$65,000 (or above $62,500 when the cost of the options is taken into account).
There is a fundamental difference between the use of forward contracts and options for hedging.Forward contracts are designed to neutralize risk by fixing the price that the hedger will pay orreceive for the underlying asset. Option contracts, by contrast, provide insurance. They offer a wayfor investors to protect themselves against adverse price movements in the future while stillallowing them to benefit from favorable price movements. Unlike forwards, options involve thepayment of an up-front fee.
Speculators
We now move on to consider how futures and options markets can be used by speculators.Whereas hedgers want to avoid an exposure to adverse movements in the price of an asset,12 CHAPTER 1
speculators wish to take a position in the market. Either they are betting that the price will go upor they are betting that it will go down.
Consider a U.S. speculator who in February thinks that the British pound will strengthenrelative to the U.S. dollar over the next two months and is prepared to back that hunch to thetune of £250,000. One thing the speculator can do is simply purchase £250,000 in the hope thatthe sterling can be sold later at a profit. The sterling once purchased would be kept in aninterest-bearing account. Another possibility is to take a long position in four CME April
futures contracts on sterling. (Each futures contract is for the Purchase of £62,500.) Supposethat the current exchange rate is 1.6470 and the April futures price is 1.6410. If the exchange
hat current exchange and the / utures price is 1.6 If the exchangerate turns out to be 1.7000 in April, the futures contract alternative enables the speculator torealize a profit of (1.7000 — 1.6410) x 250,000 = $14,750. The cash market alternative leads toan asset being purchased for 1.6470 in February and sold for 1.7000 in April, so that a profitof (1.7000 — 1.6470) x 250,000 = $13,250 is made. If the exchange rate falls to 1.6000, thefutures contract gives rise to a (1.6410 — 1.6000) x 250,000 = $10,250 loss, whereas the cashmarket alternative gives rise to a loss of (1.6470 — 1.6000) x 250,000 = $11,750. The alternativesappear to give rise to slightly different profits and losses. But these calculations do not reflectthe interest that is earned or paid. It will be shown in Chapter 3 that when the interest earnedin sterling and the interest paid in dollars are taken into account, the profit or loss from thetwo alternatives is the same.
What then is the difference between the two alternatives? The first alternative of buying sterlingrequires an up-front investment of $411,750. As we will see in Chapter 2, the second alternativerequires only a small amount of cash—perhaps $25,000-——to be deposited by the speculator inwhat is termed a margin account. The futures market allows the speculator to obtain leverage.With a relatively small initial outlay, the investor is able to take a large speculative position.
We consider next an example of how a speculator could use options. Suppose that it is Octoberand a speculator considers that Cisco is likely to increase in value over the next two months. Thestock price is currently $20, and a two-month call option with a $25 strike price is currently sellingfor $1. Table 1.2 illustrates two possible alternatives assuming that the speculator is willing toinvest $4,000. The first alternative involves the purchase of 200 shares. The second involves thepurchase of 4,000 call options (i.c., 20 call option contracts).
Suppose that the speculator’s hunch is correct and the price of Cisco’s shares rises to $35 byDecember. The first alternative of buying the stock yields a profit of
200 x ($35 — $20) = $3,000 ‘However, the second alternative is far more profitable. A call option on Cisco with a strike price
Table 1.2 Comparison of profits (losses) from two alternativestrategies for using $4,000 to speculate on Cisco stock in October
December stock price
Investor’s strategy SIS $35
 
Buy shares ($1,000) $3,000Buy call options ($4,000) $36,000Introduction 13
of $25 gives a payoff of $10, because it enables something worth $35 to be bought for $25. Thetotal payoff from the 4,000 options that are purchased under the second alternative is
4,000 x $10 = $40,000Subtracting the original cost of the options yields a net profit of$40,000 — $4,000 = $36,000
The options strategy is, therefore, 12 times as profitable as the strategy of buying the stock.Options also give rise to a greater potential loss. Suppose the stock price falls to $15 byDecember. The first alternative of buying stock yields a loss of
200 x ($20 — $15) = $1,000
Because the call options expire without being exercised, the options strategy would lead to a loss of$4,000—the original amount paid for the options.
It is clear from Table 1.2 that options like futures provide a form of leverage. For a giveninvestment, the use of options magnifies the financial consequences. Good outcomes become verygood, while bad outcomes become very bad!
Futures and options are similar instruments for speculators in that they both provide a way inwhich a type of leverage can be obtained. However, there is an important difference between the two.With futures the speculator’s potential loss as well as the potential gain is very large. With options nomatter how bad things get, the speculator’s loss is limited to the amount paid for the options.
Arbitrageurs
Arbitrageurs are a third important group of participants in futures, forward, and options markets.Arbiirage involves locking in a riskless profit by simultaneously entering into transactions iu twoor more markets. In later chapters we will see how arbitrage is sometimes possible when the futuresprice of an asset gets out of line with its cash price. We will also examine how arbitrage can be usedin options markets. This section illustrates the concept of arbitrage with a very simple example.
Consider a stock that is traded on both the New York Stock Exchange (www.nyse.com) and theLondon Stock Exchange (www.stockex.co.uk). Suppose that the stock price is $152 in New Yorkand £100 in London at a time when the exchange rate is $1.5500 per pound. An arbitrageur couldsimultaneously buy 100 shares of the stock in New York and sell them in London to obtain a risk-free profit of
100 x {($1.55 x 100) ~ $152]
or $300 in the absence of transactions costs. Transactions costs would probably eliminate the profitfor a small investor. However, a large investment house faces very low transactions costs in boththe stock market and the foreign exchange market. It would find the arbitrage opportunity veryattractive and would try to take as much advantage of it as possible.
Arbitrage opportunities such as the one just described cannot last for long. As arbitrageurs buythe stock in New York, the forces of supply and demand will cause the dollar price to rise.Similarly, as they sell the stock in London, the sterling price will be driven down. Very quickly thetwo prices will become equivalent at the current exchange rate. Indeed, the existence of profit-hungry arbitrageurs makes it unlikely that a major disparity between the sterling price and thedollar price could ever exist in the first place. Generalizing from this example, we can say that the14 CHAPTER 1
very existence of arbitrageurs means that in practice only very small arbitrage opportunities areoghecgecd than wetnne that ave cd Ge mae fnannia 1 wsarkote Te thie hagk moat of theobserved in the prices that are quoted in most financial markets. In this book most of the
arguments concerning futures prices, forward prices, and the values of option contracts will bebased on the assumption that there are no arbitrage opportunities.
1.7. OTHER DERIVATIVES
 
 
The caii and put options we have considered so far are sometimes termed “‘plain vanilla” or“standard” derivatives. Since the early 1980s, banks and other financial institutions have been veryimaginative in designing nonstandard derivatives to meet the needs of clients. Sometimes these aresold by financial institutions to their corporate clients in the over-the-counter market. On otheroccasions, they are added to bond or stock issues to make these issues more attractive to investors.Some nonstandard derivatives are simply portfolios of two or more “‘plain vanilla” call and putoptions. Others are far more complex. The possibilities for designing new interesting nonstandardderivatives seem to be almost limitless. Nonstandard derivatives are sometimes termed exoticoptions or just exotics. In Chapter 19 we discuss different types of exotics and consider how theycan he valued.
We now give examples of three derivatives that, although they appear to be complex, can bedecomposed into portfolios of plain vanilla call and put options.”
Example 1.1: Standard Oil's Bond Issue A bond issue by Standard Oil worked as follows. Theholder received no interest. At the bond’s maturity the company promised to pay $1,000 plus anadditional amount based on the price of oil at that time. The additional amount was equal to theproduct of 170 and the excess Gf any) of the price of a barrel of oil at maturity over $25, Themaximum additional amount paid was $2,550 (which corresponds to a price of $40 per barrel). Thesebonds provided holders with a stake in a commodity that was critically important to the fortunes ofthe company. If the price of the commodity went up, the company was in a good position to providethe bondholder with the additional payment.
Example 1.2: ICON In the 1980s, Bankers Trust developed index currency option notes (ICONS).These are bonds in which the amount received by the holder at maturity varies with a foreignexchange rate. Two exchange rates, K; and K>, are specified with K,; > K>. If the exchange rate at thebond’s maturity is above K,, the bondholder receives the full face value. If it is less than K», thebondholder receives nothing. Between K» and K,, a portion of the full face value is received. BankersTrust's first issue of