25, its volatility is 30% per annum, and the risk-free interest rate is10% per annum. What is the value of a nine-month European call on the futures with a strikeprice of 26?
A futures price is currently 70, its volatility is 20% per annum, and the risk-free interest rate is6% per annum. What is the value of a five-month European put on the futures with a strike priceof 65?
Suppose that a futures price is currently 35. A European call option and a European put optionon the futures with a strike price of 34 are both priced at 2 in the market. The risk-free interestrate is 10% per annum. Identify an arbitrage opportunity. Both options have one year tomaturity.
“The price of an at-the-money European call futures option always equals the price of a similarat-the-money European put futures option.’ Explain why this statement is true.
Suppose that a futures price is currently 30. The risk-free interest rate is 5% per annum. A three-month American call futures option with a strike price of 28 is worth 4. Calculate bounds for theprice of a three-month American put futures option with a strike price of 28.
oexchange rate, and the other on the dotlar—-yen exchange rate? Explain your answer.
A corporation knows that in three months it will have $5 million to invest for 90 days at LIBORminus 50 basis points and wishes to ensure that the rate obtained will be at least 6.5%. Whatposition in exchange-traded interest-rate options should the corporation take?
Prove the results in equations (13.1), (13.2), and (13.3) using Portfolios A, B, C, and D inSection 13.1.294
CHAPTER 13
ASSIGNMENT QUESTIONS
 
13.41.
13.42.
13.43.
13.44.
13.45.
13.46.
13.47.
 
 
A futures price is currently 40, It is known that at the end of three months the price will be either35 or 45, What is the value of a three-month European call option on the futures with a strikeprice of 42 if the risk-free interest rate is 7% per annum?
A stock index currently stands at 300. It is expected to increase or decrease by 10% over each ofthe next two time periods of three months. The risk-free interest rate is 8% and the dividendyield on the index is 3%. What is the value of a six-month put option on the index with a strikeprice of 300 if it is (a) European and (b) American?
Suppose that the spot price of the Canadian dollar is U.S. $0.75 and that the Canadian dollar-U.S. dollar exchange rate has a volatility of 4% per annum. The risk-free rates of interest inCanada and the United States are 9% and 7% per annum, respectively. Calculate the value of aEuropean call option to buy one Canadian dollar for U.S. $0.75 in nine months. Use put—callparity to calculate the price of a European put option to sell one Canadian dollar for U.S. $0.75in nine months. What is the price of an option to buy U.S. $0.75 with one Canadian dollar innine months?
Calculate the implied volatility of soybean futures prices from the following informationconcerning a European put on soybean futures:
Current futures price 525
Exercise price 525
Risk-free rate 6% per annumTime to maturity 5 months
Put price 20
A mutual fund announces that the salaries of its fund managers will depend on the performanceof the fund. If the fund loses money, the salaries will be zero. If the fund makes a profit, thesalaries will be proportional to the profit. Describe the salary of a fund manager as an option.How is a fund manager motivated to behave with this type of remuneration package?
Use the DerivaGem software to calculate implied volatilities for the May options on corn futuresin Table 13.5. Assume the futures prices in Table 2.2 apply and that the risk-free rate is 5% perannum. Treat the options as American and use 100 time steps. The options mature on April 21,2001. Can you draw any conclusions from the pattern of implied volatilities you obtain?
Use the DerivaGem software to calculate implied volatilities for the June 100 call and the June100 put on the Dow Jones Industrial Average in Table 13.1. The value of the DJX on March 15,2001, was 100.31. Assume the risk-free rate was 4.5%, the dividend yield was 2%. The optionsexpire on June 16, 2001. Are the quotes for the two options consistent with put—call parity?Options on Stock Indices, Currencies, and Futures 295
APPENDIX 13A
Derivation of Differential Equation Satisfied by a DerivativeDependent on a Stock Providing a Dividend Yield
Define f as the price of a derivative dependent on a stock that provides a dividend yield at rate g.We suppose that the stock price, 5, follows the process
dS = pS dt +oS dz
where dz is a Wiener process. The variables 4 and o are the expected proportional growth rate inthe stock price and the volatility of the stock price, respectively. Because the stock price provides adividend yield, 4 is only part of the expected return on the stock.*
Because f is a function of S and f, it follows from Itd’s lemma that
aeas
— (of 19S 28 afat= (Hu ust +3 oS dt + soos dz
Similarly to the procedure of Section 12.6, we can set up a portfolio consisting of
-1: derivative
+oh stockIf 11 is the value of the portfolio, thenaf=—f+—4 13A.1TI ft+ aS S ( )
and the change 311 in the value of the portfolio in a time period dt is given by equation (12.14) as_{ of  Ffagant = ( a Pages pat
In time St, the holder of the portfolio earns capital gains equal to S11 and dividends on the stockposition equal to
afqS x5 ot
Define 5W as the change in the wealth of the portfolio holder in time 6t. It follows that
_ _8f 8S, 22 afaw =( oh oso'S* +45 a (13A.2)
Because this expression is independent of the Wiener process, the portfolio is instantaneouslyriskless. HencedW = riot (13A.3)
* From equation (13.6), 4 =r—q ina risk-neutral world.296 CHAPTER 13
Substituting from equations (13A.1) and (13A.2) into equation (13A.3) gives
a ve af af( of hots +9 of) i ( s+ its) a
 
 
so that
af af 1, 200f~ get 4 1g292 ZF — 13A4a tr MS ae + 39 aa rt ( )
This is the differential equation that must be satisfied by /f.Options on Stock Indices, Currencies, and Futures 297
APPENDIX 13B
Derivation of Differential Equation Satisfied by a DerivativeDependent on a Futures Price
Suppose that the futures price F follows the processdF =F dt +0F dz (13B.1)
where dz is a Wiener process and o is constant.° Because f is a function of F and t, it follows from1té’s femma that
df= (32 pr + et Loa hor) a+ 3 oF dz (13B.2)
Consider a portfolio consisting of
—I: derivative
af
top : futures contracts
Define I] as the value of the portfolio and let 51], &f, and 3F be the change in I, f, and F,respectively, in time 5t. Because it costs nothing to enter into a futures contract,
n=-f (13B.3)
In a time period dt, the holder of the portfolio earns capital gains equal to —df from the derivativeand income of
af
—5F
oFfrom the futures contract. Define dW as the total change in wealth of the portfolio holder in timebt. It follows that
aw = ol sp af
The discrete versions of equations (13B.1) and (13B.2) are
OF = pF dt + oF 8z
éf= (Gee Freitas fe iB thot) + thors
where 6z = ¢ /ét and ¢ is a random sample from a standardized normal distribution. It followsthat
éWw= op aR or jor (13B.4)
5 From the arguments in Section 13.7, 4 = 0 in a risk-neutral world.298 CHAPTER 13
This is riskless. Hence it must also be true that5W = rll dt (13B.5)If we substitute for 1 from equation (13B.3), then equations (13B.4) and (13B:5) give
oe yas f aera —rf &tHenceart Ltt ar =rfThis has the same form as equation (13A.4) with gq set equal to r. We deduce that a futures price
can be treated in the same way as a stock providing a dividend yield at rate r for the purpose ofvaluing derivatives. 
CHAPTER 14THE GREEK LETTERS
A financial institution that sells an option to aclient in the over-the-counter markets is faced with theproblem of managing its risk. If the option happens to be the same as one that is traded on anexchange, the financial institution can neutralize its exposure by buying on the exchange the sameoption as it has sold. But when the option has been tailored to the needs of a client and does notcorrespond to the standard products traded by exchanges, hedging the exposure is far more difficult.
In this chapter we discuss some of the alternative approaches to this problem. We cover what arecommonly referred to as the “Greek letters” or simply the “Greeks.” Each Greek letter measures adifferent dimension to the risk in an option position and the aim of a trader is to manage theGreeks so that all risks are acceptable. The analysis presented in this chapter is applicable tomarket makers in options on an exchange as well as to traders working for financial institutions.
Toward the end of the chapter we will consider the creation of options synthetically. This turnsout to be very closely related to the hedging of options. Creating an option position synthetically isessentially the same task as hedging the opposite option position. For example, creating a long calloption synthetically is the same as hedging a short position in the call option.
14.1 ILLUSTRATION
 
 
In the next few sections we use as an example the position of a financial institution that has soldfor $300,000 a European call option on 100,000 shares of a non-dividend-paying stock. We assumethat the stock price is $49, the strike price is $50, the risk-free interest rate is 5% per annum, thestock price volatility is 20% per annum, the time to maturity is 20 weeks (0.3846 years), and theexpected return from the stock is 13% per annum.! With our usual notation, this means that
So=49, K=50, r=0.05, o=0.20, T=0.3846, w=0.13
The Black-Scholes price of the option is about $240,000. The financial institution has thereforesold the option for $60,000 more than its theoretical value. But it is faced with the problem ofhedging the risks.”
' As shown in Chapters 10 and 12, the expected return is irrelevant to the pricing of an option. lt is given herebecause it can have some bearing on the effectiveness of a hedging scheme.
? Financial institutions do not normally write call options on individual stocks. However, a call option on a stock isa convenient example with which to develop our ideas. The points that will be made apply to other types of optionsand to other derivatives.
299300 CHAPTER 14
14.2 NAKED AND COVERED POSITIONS
 
 
One strategy open to the financial institution is to do nothing. This is sometimes referred to asadopting a naked position. \ is a strategy that works well if the stock price is below 550 at the endof the 20 weeks. The option then costs the financial institution nothing and it makes a profit of$300,000. A naked position works less well if the call is exercised because the financial institutionthen has to buy-100,000 shares at the market price prevailing in 20 weeks to cover the call. The costto the financial institution is 100,000 times the amount by which the stock price exceeds the strikeprice. For example, if after 20 weeks the stock price is $60, the option costs the financial institution$1,000,000. This is considerably greater than the $300,000 premium received.
As an alternative to a naked position, the financial institution can adopt a covered position. Thisinvolves buying 100,000 shares as soon as the option has been sold. If the option is exercised, thisstrategy works well, but in other circumstances it could lead to a significant loss, For example, ifthe stock price drops to $40, the financial institution loses $900,000 on its stock position. This isconsiderably greater than the $300,000 charged for the option.?
Neither a naked position nor a covered position provides a satisfactory hedge. If the assumptionsunderlying the Black-Scholes formula hold, the cost to the financial institution should always be
2740. 000 on avernze for both anproaches.* But on any one occasion the cost is Hable to ranee frombo4U,UUU Of average ior OOtn approacnes. GUL ON any One Occasion tne COst iS aoe tO range irom
zero to over $1,000,000. A perfect hedge would ensure that the cost is always $240,000. For aperfect hedge, the standard deviation of the cost of writing and hedging the option is zero,
14.3, A STOP-LOSS STRATEGY
 
One interesting hedging scheme that is sometimes proposed involves a stop-loss strategy. Toillustrate the basic idea, consider an institution that has written a call option with strike price Kto buy one unit of a stock. The hedging scheme involves buying one unit of the stock as soon as itsprice rises above K and selling it as soon as its price falls below K. The objective is to hold a nakedposition whenever the stock price is less than K and a covered position whenever the stock price isgreater than K. The scheme is designed to ensure that at time 7 the institution owns the stock if theoption closes in the money and does not own it if the option closes out of the money. The strategyappears to produce payoffs that are the same as the payoffs on the option. In the situationillustrated in Figure 14.1, it involves buying the stock at time 1,, selling it at time tf, buying itat time ft, selling it at time t,, buying it at time t;, and delivering it at time T.
As usual, we denote the initial stock price by Sp. The cost of setting up the hedge initially is Spif Sy > K and zero otherwise. It seems as though the total cost, Q, of writing and hedging theoption is the option’s intrinsic value, that is,
QO = max(Sy — K, 0) (4.1)
This is because all purchases and sales subsequent to time zero are made at price K. If this were infact correct, the hedging scheme would work perfectly in the absence of transactions costs.
3 Put-call parity shows that the exposure from writing a covered call is the same as the exposure from writing anaked put.
* More precisely, the present value of the expected cost is $240,000 for both approaches, assuming that appropriaterisk-adjusted discount rates are used.The Greek Letters 301
4 Stockprice, S(4)
 
 
Lt t 1 1 —L. L —>ty tk ty t5 T
Figure 14.1 A stop-loss strategy
Furthermore, the cost of hedging the option would always be fess than its Black-Scholes price.Thus, an investor could earn riskless profits by writing options and hedging them.
There are two basic reasons why equation (14.1) is incorrect. The first is that the cash flows to thehedger occur at different times and must be discounted. The second is that purchases and salescannot be made at exactly the same price K. This second point is critical. If we assume a risk-neutralworld with zero interest rates, we can justify ignoring the time value of money. But we cannotlegitimately assume that both purchases and sales are made at the same price. If markets are efficient,the hedger cannot know whether, when the stock price equals K, it will continue above or below K.
As a practical matter, purchases must be made at a price K + € and sales must be made at a priceK — , for some small positive number ¢. Thus, every purchase and subsequent sale involves a cost(apart from transaction costs) of 2e. A natural response on the part of the hedger is to monitorprice movements more closely so that € is reduced. Assuming that stock prices change continu-ously, € can be made arbitrarily small by monitoring the stock prices closely. But as € is madesmaller, trades tend to occur more frequently. Thus, even though the cost per trade is reduced, it isoffset by the increasing frequency of trading. As « > 0, the expected number of trades tends toinfinity.°
A stop-loss strategy, although superficially attractive, does not work particularly well as ahedging scheme. Consider its use for an out-of-the-money option. If the stock price never reachesthe strike price of K, the hedging scheme costs nothing. If the path of the stock price crosses thestrike price level many times, the scheme is quite expensive. Monte Carlo simulation can be used toassess the overall performance of stop-loss hedging. This involves randomly sampling paths for thestock price and observing the results of using the scheme. Table 14.1 shows the results for theoption considered earlier. It assumes that the stock price is-observed at the end of time intervals of
-
a
5 As mentioned in Section 11.2, the expected number of tittiés a Wiener process equals any particular value in a giventime interval is infinite.302 - CHAPTER 14
 
Table 14.1 Performance of stop-loss strategy. (The performance measureis the ratio of the standard deviation of the cost of writing the option andhedging it to the theoretical price of the option.)
bt (weeks) 5 4 2 1 0.5 0.25Hedge performance 1.02 0.93 0.82 0.77 0.76 0.76
length 5t.° The hedge performance measure is the ratio of the standard deviation of the cost ofhedging the option to the Black-Scholes price of the option. Each result is based on 1,000 samplepaths for the stock price and has a standard error of about 2%. It appears to be impossible toproduce a value for the hedge performance measure below 0.70 regardless of how small dr is made.
14.4 DELTA HEDGING
 
 
Most traders use more sophisticated hedging schemes than those mentioned so far. These involvecalculating measures such as delta, gamma, and vega. Here we consider the role played by delta.The de/ta of an option, A, was introduced in Chapter 10. It is defined as the rate of change of theoption price with respect to the price of the underlying asset. It is the slope of the curve that relatesthe option price to the underlying assct price. Suppose that the delta of a call option on a stock is 0.6.This means that when the stock price changes by a small amount, the option price changes by about60% of that amount. Figure 14.2 shows the relationship between a call price and the underlyingstock price. When the stock price corresponds to point A, the option price corresponds to point B,and A is the slope of the line indicated. In general,oc
A=55
where c is the price of the call option and S is the stock price.
Optionprice
        
Slope =A=0.6
Stock
I\!I price
 
A
Figure 14.2 Calculation of delta
® The precise hedging rule used was as follows. If the stock price moves from below K to above K in a time intervalof length 7, it is bought at the end of the interval. If it moves from above K to below K in the time interval, it is soldat the end of the interval. Otherwise, no action is taken.The Greek Letters 303
Suppose that in Figure 14.2 the stock price is $100 and the option price is $10. Imagine aninvestor who has sold 20 call option contracts—that is, options to buy 2,000 shares. The investor’sposition could be hedged by buying 0.6 x 2,000 = 1,200 shares. The gain (loss) on the optionposition would then tend to be offset by the loss (gain) on the stock position. For example, if thestock price goes up by $1 (producing a gain of $1,200 on the shares purchased), the option pricewill tend to go up by 0.6 x $1 = $0.60 (producing a loss of $1,200 on the options written); if thestock price goes down by $1 (producing a loss of $1,200 on the shares purchased), the option pricewill tend to go down by $0.60 (producing a gain of $1,200 on the options written).
In this example, the delta of the investor’s option position is 0.6 x (—2,000) = —1,200. In otherwords, the investor loses 1,200 8S on the short option position when the stock price increases by 48S.The delta of the stock is 1.0, and the long position in 1,200 shares has a delta of +1,200. The deltaof the investor’s overall position is therefore zero. The delta of the stock position offsets the delta ofthe option position. A position with a delta of zero is referred to as being delta neutral.
It is important to realize that, because delta changes, the investor’s position remains deltahedged (or delta neutral) only for a relatively short period of time. The hedge has to be adjusted_ periodically. This is known as rebalancing. In our example, at the end of three days the stock pricemight increase to $110. As indicated by Figure 14.2, an increase in the stock price leads to anincrease in delta. Suppose that delta rises from 0.60 to 0.65. An extra 0.05 x 2,000 = 100 shareswould then have to be purchased to maintain the hedge. The delta-hedging scheme just described isan example of a dynamic-hedging scheme. It can be contrasted with static-hedging schemes, wherethe hedge is set up initially and never adjusted. Static hedging schemes are sometimes also referredto as hedge-and-forget schemes.
Delta is closely related to the Black-Scholes—Merton analysis. As explained in Chapter 12, Black,Scholes, and Merton showed that it is possible to set up a riskless portfolio consisting of a position inan option on a stock and a position in the stock. Expressed in terms of A, the Black-Scholesportfolio is
—l: option
+A: shares of the stock
Using our new terminology, we can say that Black and Scholes valued options by setting up adelta-neutral position and arguing that the return on the position should be the risk-free interestrate.Delta of European Stock OptionsFor a European call option on a non-dividend-paying stock, it can be shown that
A= N(d\)where d, is defined as in equation (12.20). Using delta hedging for a short position in a Europeancall option therefore involves keeping a long position of N(d,) shares at any given time. Similarly,using delta hedging for a long position in a European call option involves maintaining a short
position of N(d,) shares at any given time.For a European put-opition on a non-dividend-paying stock, delta is given by
A=MN(d,)-1
Delta is negative, which means that a long position in a put option should be hedged with a long304 CHAPTER 14
Delta of Delta ofcall put
Stock price0.0
     
Stock price
0.0 -1.0
(a) (b)
Figure 14.3 Variation of delta with stock price: (a) call option and (b) put option on anon-dividend-paying stock
position in the underlying stock, and a short position in a put option should be hedged with ashort position in the underlying stock. Figure 14.3 shows the variation of the delta of a call optionand a put option with the stock price. Figure 14.4 shows the variation of delta with the time tomaturity for in-the-money, at-the-money, and out-of-the-money call options.
Delta of Other European Options
For European call options on an asset paying a yield g, we have
A=e Nd)
A Delta
In the money
           
At the money
Out of the money
 
Time to expiration
Figure 14.4 Typical patterns for variation of delta with time to maturity for a call optionThe Greek Letters 305
 
where d, is defined by equation (13.4). For European put options on the asset, we haveA= (N(d) ~ 1)
When the asset is a stock index, these formulas are correct with g equal to the dividend yield on theindex. When the asset is a currency, they are correct with q equal to the foreign risk-free rate, ry.
Whee the aecet res contract eorrect Wy — Anas ectic cok free x:
When the asset is a futures coniract, they are correct with gq equal to the domestic risk-free rate, 7.
Example 14.1. A U.S. bank has sold six-month put options on £1 million with a strike priceof 1.6000 and wishes to make its portfolio delta neutral. Suppose that the current exchange rate is1.6200, the risk-free interest rate in the United Kingdom is 13% per annum, the risk-free interest ratein the United States is 10% per annum, and the volatility of sterling is 15%. In this case, Sy = 1.6200,K = 1.6000, r = 0.10, r; = 0.13, o = 0.15, and T = 0.5. The delta of a put option on a currency is
[M(d,) — Ler"where d, is given by equation (13.9). It can be shown thatd, =0.0287 and M(d)) = 0.5115
so that the delta of the put option is —0.458. This is the delta of a long position in one put option.(It means that when the exchange rate increases by 8S, the price of the put goes down by 45.8% of 55.)The delta of the bank’s total short option position is +458,000. To make the position delta neutral,we must therefore add a short sterling position of £458,000 to the option position. This short sterlingposition has a delta of —458,000 and neutralizes the delta of the option position.
Delta of Forward Contracts
The concept of delta can be applied to financial instruments other than options. Consider a forwardcontract on a non-dividend-paying stock. Equation (3.9) shows that the value of a forward contractis Sy — Ke~"’, where K is the delivery price and T is the forward contract’s time to maturity. Whenthe price of the stock changes by 6S, with all else remaining the same, the value of a forward contracton the stock also changes by 6S. The delta of a forward contract on one share of the stock is thereforealways 1.0. This means that a short forward contract on one share can be hedged by purchasing oneshare; a long forward contract on one share can be hedged by shorting one share.
For an asset providing a dividend yield at rate g, equation (3.11) shows that the forwardcontract’s delta is e~?”. In the case of a stock index, g is set equal to the dividend yield on theindex. For a currency, it is set equal to the forcign risk-free rate, ry.
Delta of a Futures Contract
From equation (3.5), the futures price for a contract on a non-dividend-paying stock is Sge’’,
where T is the time to maturity of the futures contract. This shows that when the price of the stock
changes by 5S, with all else remaining the same, the futures price changes by 65 e”. Because futurescontracts are marked to market daily, the holder of a long futures contract makes an almostimmediate gain of this amount. The delta of a futures contract is therefore e’”. For a futurescontract on an asset providing a dividend yield at rate g, equation (3.7) shows similarly that deltaise". {1 is interesting that the impact of marking to market is to make the deltas of futures and
7 These are hedge-and-forget schemes. Because delta is always 1.0, no changes need to be made to the position in thestock during the life of the contract.306 CHAPTER 14
forward contracts slightly different. This is true even when interest rates are constant and theforward price equals the futures price.Sometimes a futures contract is used to achieve a delta-neutral position. Define:
T: Maturity of futures contractH,: Required position in asset for delta hedgingH,: Alternative required position in futures contracts for delta hedging
Tf the underlying asset is a non-dividend-paying stock, the analysis we have just given shows thatHp =e Hy, (14.2)
When the underlying asset pays a dividend yield g, we haveHp =e" H, (14.3)
For a stock index we set g equal to the dividend yield on the index; for a currency we set it equal tothe foreign risk-free rate, rf, 80 thatAH, =e A, (14.4)
Example 14.2 Consider the option in the previous example where hedging using the currencyrequires a short position of £458,000. From equation (14.4), hedging using nine-month currencyfutures requires a short futures position of
¢7 (0-10-0.13)x9/12 458 Q00
or £468,442. Because each futures contract is for the purchase or sale of £62,500, seven contractspeer ire ae AGB dD /E9 SOO)SHOW DE SOTTO (SCVCTL 15 LC Licarest WOE NUDE LO 400,442/02,90U).
Dynamic Aspects of Delta Hedging
Tables 14.2 and 14.3 provide two examples of the operation of delta hedging for the example inSection 14.1, The hedge is assumed to be adjusted or rebalanced weekly. The initial value of deltacan be calculated from the data in Section 14.1 as 0.522. This means that as soon as the option iswritten, $2,557,800 must be borrowed to buy 52,200 shares at a price of $49. The rate of interest is5%. An interest cost of $2,500 is therfore incurred in the first week.
In Table 14