itates hedging, arbitrage,tion. It also tends to make the markets more cfficicat.
A final point is that futures options tend to entail lower transactions costsmany situations.
   
 
Put-Call Parity
In Chapter 8 we derived a put-call parity relationship for European stock options. We now present \a similar argument to derive a put-call parity relationship for European futures options on theassumption that there is no difference between the payoffs from futures and forward contracts.Consider European call and put futures options, both with/strike price K and time to expiration 7.We can form two portfolios:
    
Portfolio A: a European call futures option plus an amount of cash equal to Ke”
Portfolio B: a European put futures option plus a long futures contract plus an amount of cashequal to Fye”
In portfolio A the cash can be invested at the risk-free rate, r, and\will grow to K at time T. Let Fr bethe futures price at maturity of the option. If F; > K, the call option in portfolio A is exercised andportfolio A is worth F;. If F; < K, the call is not exercised and portfolio-A is worth K. The value ofportfolio A at time T is therefore
 
max(Fr, K) ae
In portfolio B the cash can be invested at the risk-free rate to grow to Fy at time T. The put optionprovides a payoff of max(K — Fr, 0). The futures contract provides a payoff of F; — Fy. The valueof portfolio B at time T is therefore
Fo + (Fr — Fo) + max(K — Fp, 0) = max(Fr, K)
a eo284 CHAPTER 13
Because the two portfolios have the same value at time T and there are no early exerciseopportunities, it follows that they are worth the same today. The value of portfolio A today is
c+Ke™
where c is the price“of the ca ures option. The marking-to-market process ensures that thefutures cefitract in portfolio B is worthrzero today. Therefore portfolio B is worth4
pepe
where p is the price of the put futures option. Hen
                
rT
c+ Ke = p+ het (13.13)
Example13.9 Suppose that the price of a European call option on silver futures for delivery in sixmonths is $0.56 per ounce when the exercise price is $8.50. Assume that the silver futures price fordelivery in six months is currently $8.00 and the risk-free interest rate for an investment that maturesin six months is 10% per annum. From a rearrangement ofjequation (13.13), the price of a Europeanput option on silver futures with the same maturity and exercise date as the call option is
0.56 + 8.500% 95 _ 8.09679 *95 = 1.04
For American futures the put—call parity relationships (see Problem 13.20)
13.6 VALUATIONBINOMIAL
  
RE
 
 
ey difference between futures options and stock options is that there are no up-front costswhen a futures contract is entered into. ee
Suppose that the current futures price is 30 and it is expected to move either up to 33 or down to 28over the next month. We consider a one-month call option on the futures with a strike price of 29 andignore daily settlement. The situation is shown in Figure 13.2. If the futures price proves to be 33, thepayoff from the option is 4 and the value of the futures contract is 3. If the futures price proves tobe 28, the payoff from the option is zero and the value of the futures contract is —2.
33
30
28
Figure 13.2 Futures price movements in numerical exampleOptions on Stock Indices, Currencies, and Futures 285
To set up a riskless hedge, we consider a portfolio consisting of a short position in one optioncontract and a long position in A futures contracts. If the futures price moves up to 33, the value ofthe portfolio is 3A — 4; if it moves down to 28, the value of the portfolio is -2A. The portfolio istiskless when these are the same—that is, when
3A -4=-2Aor A= 0.8.For this value of A, we know the portfolio will be worth 3 x 0.8 —4=—1.6 in one month.Assume a risk-free interest rate of 6%. The value of the portfolio today must be
1,627 9.06% 0.08333 = —1.592
The portfolio consists of one short option and A futures contracts. Because the value of the futurescontract today is zero, the value of the option today must be 1.592.
A Generalization
We can generalize this analysis by considering a futures price that starts at Fy and is anticipated torise to Fou or move down to Fod over the time period T. We consider a derivative maturing at theend of the time period, and we suppose that its payoff is f, if the futures price moves up and f, if itmoves down. The situation is summarized in Figure 13.3.
The riskless portfolio in this case consists of a short position in one option combined with a longposition in A futures contracts, where
fun fa
= Fou — FodThe value of the portfolio at the end of the time period, then, is always(Fou — Fo)A — fuDenoting the risk-free interest rate by r, we obtain the value of the portfolio today as[(Fou — Fo) A — fle”
Another expression for the present value of the portfolio is —f, where f is the value of the optiontoday. It follows that~f = [Fou — FA — file
_ FouSuFofFodfa
Figure 13.3 Futures price and option price in general situation286 CHAPTER 13
 
Substituting for A and simplifying reduces this equation to
 
f=e "ph tC - pf (13.14)where—dp= . a (13.15)
In the numerical example in Figure 13.2, u = 1.1, d= 0.9333, r= 0.06, T = 0.08333, f, =4,
and f; = 0. From equation (13.15),1— 0.9333
P=Ti 09333 94
and, from equation (13.14),f = 7 00X0.0833319 4 x 4 +0.6 x 0] = 1.592
This result agrees with the answer obtained earlier for this example.
1
wdaai0Tlnuv0om>
vu
 
There is a general result that makes the analysis of futures options analogous to the analysis ofoptions on a stock paying a dividend yield. This result is that a futures price behaves in the sameway as a stock paying a dividend yield at the domestic risk-free interest rate, r.
One clue that this might be so is given by comparing equations (13.14) and (13.15) withequations (13.7) and (13.8). The two sets of equations are identical when we set g =r. Anotherclue is that the put-call parity relationship for futures options prices are the same as those foroptions on a stock paying a dividend yield at rate g when the stock price is replaced by the futuresprice and g =r.
We can understand the general result by noting that a futures contract requires zero investment.In a risk-neutral world the expected profit from holding a position in an investment that costs zeroto set up must be zero. The expected payoff from a futures contract in a risk-neutral world musttherefore be zero. It follows that the expected growth rate of the futures price in a risk-neutralworld must be zero. As pointed out in Section 13,2, a stock paying a dividend at rate g grows at anexpected rate of r — g in a risk-neutral world. If we set g = r, the expected growth rate of the stockprice is zero, making it analogous to a futures price.
To prove the result formally, we derive in Appendix 13B the differential equation for a derivativedependent on a futures price. This differential equation is the same as that in Appendix 13A foraderivative dependent on a stock providing a known dividend yield, g, equal to the risk-free interestrate, r.
The Expected Growth Rate of a Futures Price
The result, that the expected growth rate in a futures price in a risk-neutral world is zero, is a verygeneral one. It is true for all futures prices. It applies in the world where interest rates are stochasticas well as the world where they are constant.”
> Note that the result applies in what we will refer to as the sraditional risk-neutral world. In later chapters we willconsider other risk-neutral worlds.Options on Stock indices, Currencies, and Futures 287
Because the expected growth rate of the futures price is zero,Fy = E(Fr)
where Fr is the futures price at the maturity of the contract, Fo is the futures price at time zero, and
E denotes the expected value in a risk-ncutral world. Because where S. is the soot priceE denotes the expected value in a risk-neutral world. Because Fp = S;, where Sz is the spot price at
time T, it follows thatFy = E(S;) (13.16)
This means that for all assets the futures price equals the expected future spot price in a risk-neutral world.
13.8. BLACK’S MODEL FOR VALUING FUTURES OPTIONS
 
European futures options can be valued by extending the results we have produced. Fischer Blackwas the first to show this in a paper published in 1976.> The underlying assumption is thatfutures prices have the same lognormal property that we assumed for stock prices in Chapter 12.The European call price, c, and the European put price, p, for a futures option are given byequations (13.4) and (13.5) with Sp replaced by Fy and g=r:
c =e" [FyN(d\) — KN(d))] (13.17)p=e "[KN(—dy) — FyM(—4,)] (13.18)wherea= In(Fo/K) + 0° T/21= oVT—_ In(fo/K)— 7/2 _ _= oF =d, oJVT
and o is the volatility of the futures price. When the cost of carry and the convenience yield arefunctions only of time, it can be shown that the volatility of the futures price is the same as thevolatility of the underlying asset. Note that Black’s model does not require the options contractand the futures contract to mature at the same time.
Example 13.10 Consider a European put futures option on crude oil. The time to the option’smaturity is four months, the current futures price is $20, the exercise price is $20, the risk-free interestrate is 9% per annum, and the volatility of the futures price is 25% per annum. In this case Fy = 20,K =20, r= 0.09, T = 4/12, o = 0.25, and In(Fy/K) = 0, so that
dy = = _oonie, , = - 227 - -oo7216
N(—d,) = 0.4712, | N(—dy) = 0.5288
3 See F. Black, “The Pricing of Commodity Contracts,” Journal of Financial Economics, 3 (March 1976), 167-79.288 CHAPTER 13
and the put price p is given by
p =e 4/2099 x 0.5288 — 20 x 0.4712) = 1.12or $1.12.
13.9 FUTURES OPTIONS vs. SPOT OPTIONS
 
 
 
In this section we compare options on futures and options on spot when they have the same strikeprice and time to maturity. An option on a spot, or spot option, is a regular option to buy or sell theunderlying asset in the spot market.
The payoff from a European spot call option with strike price K is
max(S; — K, 0)
where S; is the spot price at the option’s maturity. The payoff from a European futures call option
with the same strike price ismax(F; — K, 0)
where F; is the futures price at the option’s maturity. If the European futures option matures at thesame time as the futures contract, F; = S; and the two options are in theory equivalent. If theEuropean call futures option matures before the futures contract, it is worth more than thecorresponding spot option in a normal market (where futures prices are higher than spot prices)and less than the corresponding spot option in an inverted market (where futures prices are lowerthan spot prices).
Similarly, a European futures put option is worth the same as its spot option counterpart when
the futures option matures at the same time as the futures contract. If the European put futuresoption matures before the futures contract, it is worth less than the corresponding spot option in a
normal market and more than the corresponding spot option in an inverted market.
Results for American Options
Traded futures options are, in practice, usually American. If we assume that the risk-free rate ofinterest, r, is positive, then there is always some chance that it will be optimal to exercise an Americanfutures option early. American futures options are, therefore, worth more than their Europeancounterparts. We will look at numerical procedures for valuing futures options in Chapter 18.
It is not generally true that an American futures option is worth the same as the correspondingAmerican spot option when the futures and options contracts have the same maturity. Suppose,for example, that there is a normal market with futures prices consistently higher than spot pricesprior to maturity. This is the case with most stock indices, gold, silver, low-interest currencies, andsome commodities. An American call futures option must be worth more than the correspondingAmerican spot call option. The reason is that in some situations the futures option will be exercisedearly, in which case it will provide a greater profit to the holder. Similarly, an American put futuresoption must be worth less than the corresponding American spot put option. If there is an invertedmarket with futures prices consistently lower than spot prices, as is the case with high-interestcurrencies and some commodities, the reverse must be true. American call futures options areworth less than the corresponding American spot call option, whereas American put futuresoptions are worth more than the corresponding American spot put option.Options on Stock indices, Currencies, and Futures 289
 
The differences just described between American futures options and American spot optionshold true when the futures contract expires later than the options contract, as well as when the twoexpire at the same time. In fact, the later the futures contract expires, the greater the differencestend to be.
SUMMARY
 
 
The Black-Scholes formula for valuing European options on a non-dividend-paying stock can beextended to cover European options on a stock providing a known dividend yield. In practice,stocks do not provide known dividend yields. However, a number of other assets on which optionsare written can be considered to be analogous to a stock providing a dividend yield. In particular,
\. An index is analogous to a stock providing a dividend yield. The dividend yield is the averagedividend yield on the stocks composing the index.
2. A foreign currency is analogous to a stock providing a dividend yield where the dividendyield is the foreign risk-free interest rate.
3. A futures price is analogous to a stock providing a dividend yield where the dividend yield isequal to the domestic risk-free interest rate.
The extension to Black-Scholes can therefore be used to value European options on indices,foreign currencies, and futures contracts. As we will see in Chapter 18, these analogies are alsouseful in valuing numerically American options on indices, currencies, and futures contracts.
Index options are settled in cash. Upon exercise of an index call option, the holder receives theamount by which the index exceeds the strike price at close of trading. Similarly, upon exercise ofan index put option, the holder receives the amount by which the strike price exceeds the index atclose of trading. Index options can be used for portfolio insurance. If the portfolio has a f of 1.0, itis appropriate to buy one put option for each 100Sp dollars in the portfolio, where Sp is the value ofthe index; otherwise, 8 put options should be purchased for each 1005S) dollars in the portfolio,where f is the beta of the portfolio calculated using the capital asset pricing model. The strike priceof the put options purchased should reflect the level of insurance required.
Currency options are traded both on organized exchanges and over the counter. They can beused by corporate treasurers to hedge foreign exchange exposure. For example, a U.S. corporatetreasurer who knows that sterling will be received at a certain time in the future can hedge bybuying put options that mature at that time. Similarly, a U.S. corporate treasurer who knows thatsterling will be paid at a certain time in the future can hedge by buying call options that mature atthat time.
Futures options require the delivery of the underlying futures contract upon exercise. When acall is exercised, the holder acquires a long futures position plus a cash amount equal to theexcess of the futures price over the strike price. Similarly, when a put is exercised, the holderacquires a short position plus a cash amount equal to the excess of the strike price over thefutures price. The futures contract that is delivered typically expires slightly later than the option.If we assume that the two expiration dates are the same, a European futures option is worthexactly the same as the corresponding European spot option. However, this is not true ofAmerican options. If the futures market is normal, an American call futures option is worthmore than the corresponding American spot call option, while an American put futures option isworth less than the corresponding American spot put option. If the futures market is inverted.the reverse is true.290 CHAPTER 13
SUGGESTIONS FOR FURTHER READING
 
General
Merton, R. C., “Theory of Rational Option Pricing,” Bell Journal of Economics and ManagementScience, 4 (Spring 1973), 141-83.
Stoll, H. R., and R. E. Whaley, “New Option Instruments: Arbitrageable Linkages and Valuation,”Advances in Futures and Options Research, 1, pt. A (1986), 25-62.
On options on stock indices
Chance, D. M., “Empirical Tests of the Pricing of Index Call Options,” Advances in Futures and OptionsResearch, 1, pt. A (1986), 141-66,
On options on currencies
Amin, K., and R. A. Jarrow, “Pricing Foreign Currency Options under Stochastic Interest Rates,”Journal of International Money and Finance, 10 (1991), 310-29.
Biger, N., and J. Hull, “The Valuation of Currency Options,” Financial Management, 12 (Spring 1983),24-28.
Bodurtha, J. N., and G. R. Courtadon, “‘Tests of an American Option Pricing Model on the ForeignCurrency Options Market,” Journal of Financial and Quantitative Analysis, 22 (June 1987), 153-67.
Garman, M. B., and S. W. Kohlhagen, “Foreign Currency Option Values,” Journal of InternationalMoney and Finance, 2 (December 1983), 231-37.
Grabbe, J. O., “The Pricing of Call and Put Options on Foreign Exchange,” Journal of InternationalMoney and Finance, 2 (December 1983), 239-53.
On options on futures
Black, F., “The Pricing of Commodity Contracts,” Journal of Financial Economics, 3 (March 1976),167-79.
Brenner, M., G. Courtadon, and M. Subrahmanyam, “Options on the Spot and Options on Futures,”Journal of Finance, 40 (December 1985), 1303-17.
Hilliard, J. E., and J. Reis, “Valuation of Commodity Futures and Options under StochasticConvenience Yields, Interest Rates, and Jump Diffusions in the Spot,” Journal of Financial andQuantitative Analysis, 33, no. 1 (March 1998), 61-86.
Miltersen, K. R., and E. 8. Schwartz, “Pricing of Options on Commodity Futures with Stochastic TermStructures of Convenience Yields and Interest Rates,” Journal of Financial and Quantitative Analysis,33, no. 1 (March 1998), 33-59.
Ramaswamy, K., and S. M. Sundaresan, “The Valuation of Options on Futures Contracts,”” Journal ofFinance, 40 (December 1985), 1319-40.
Wolf, A., “Fundamentals of Commodity Options on Futures,” Journal of Futures Markets, 2 (1982),391-408.Options on Stock Indices, Currencies, and Futures 291
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
13.1.
13,2,
13.3.
13.4,
13.5,13.6.13.7,
13.8.
13.9.
13.10.13.11.13.12.
13.13.
13.14.
13.15.
13.16.13.17.
 
A portfolio is currently worth $10 million and has a beta of 1.0. The S&P 100 is currentlystanding at 500. Explain how a put option on the S&P 100 with a strike of 480 can be used toprovide portfolio insurance.
“Once we know how to value options on a stock paying a dividend yield, we know how to valueoptions on stock indices, currencies, and futures.” Explain this statement.
A stock index is currently 300, the dividend yield on the index is 3% per annum, and the risk-freeinterest rate is 8% per annum. What is a lower bound for the price of a six-month European calloption on the index when the strike price is 290?
A currency is currently worth $0.80. Over each of the next two months it is expected toincrease or decrease in value by 2%. The domestic and foreign risk-free interest rates are 6%and 8%, respectively. What is the value of a two-month European call option with a strikeprice of $0.80?
Explain the difference between a call option on yen and a call option on yen futures.Explain how currency options can be used for hedging.
Calculate the value of a three-month at-the-money European call option on a stock index whenthe index is at 250, the risk-free interest rate is 10% per annum, the volatility of the index is18% per annum, and the dividend yield on the index is 3% per annum.
Consider an American call futures option where the futures contract and the option contractexpire at the same time. Under what circumstances is the futures option worth more than thecorresponding American option on the underlying asset?
Calculate the value of an eight-month European put option on a currency with a strike priceof 0.50. The current exchange rate is 0.52, the volatility of the exchange rate is 12%, thedomestic risk-free interest rate is 4% per annum, and the foreign risk-free interest rate is8% per annum.
Why are options on bond futures more actively traded than options on bonds?
“A futures price is like a stock paying a dividend yield.” What is the dividend yield?
A futures price is currently 50. At the end of six months it will be either 56 or 46. The risk-freeinterest rate is 6% per annum. What is the value of a six-month European call option with astrike price of 50?
Calculate the value of a five-month European put futures option when the futures price is $19,the strike price is $20, the risk-free interest rate is 12% per annum, and the volatility of thefutures price is 20% per annum.
A total return index tracks the return, including dividends, on a certain portfolio. Explain howvou would valuc (a) forward contracts and (6) Eurspean ontions on the index
you would valuc {a} forward contracts and (b) European options on the index.
The S&P 100 index currently stands at 696 and has a volatility of 30% per annum. The risk-freerate of interest is 7% per annum and the index provides a dividend yield of 4% per annum.Calculate the value of a three-month European put with strike price 700.
What is the put-call parity relationship for European currency options?
A foreign currency is currently worth $1.50. The domestic and foreign risk-free interest rates are5% and 9%, respectively. Calculate a lower bound for the value of a six-month call option onthe currency with a strike price of $1.40 if it is (a) European and (b) American.292
CHAPTER 13
 
13.18.
13.19,
13.20.
13.21.
13.22.
13.23.
13.24.
13.25.
13.26.
Consider a stock index currently standing at 250. The dividend yield on the index is 4% peratinum and the risk-free rate is 6% per auiuin, A three-mouth European call option ou the indexwith a strike price of 245 is currently worth $10. What is the value of a three-month Europeanput option on the index with a strike price of 245?
Show that if C is the price of an American call with strike price K and maturity T on a stockproviding a dividend yield of g, and P is the price of an American put on the same stock with thesame strike price and exercise date, then
Soe" -K<C-P<S)—Ke™
where Spo is the stock price, r is the risk-free interest rate, and r > 0. (Hint: To obtain the first halfof the inequality, consider possible values of:Portfolio A: a European call option plus an amount K invested at the risk-free rate;Portfolio B: an American put option plus e~” of stock, with dividends being reinvested in thestock.To obtain the second half of the inequality, consider possible values of:Portfolio C: an American call option plus an amount Ke~"?~ invested at the risk-free rate;Portfolio D: a European put option plus one stock, with dividends being reinvested in thestock.) ,
Show that if C is the price of an American call option on a futures contract when the strike priceis K and the maturity is T, and P is the price of an American put on the same futures contractwith the same strike price and exercise date, then
Foe? —K << C—-P<F)-Ke™where Fy is the futures price and r is the risk-free rate. Assume that r > 0 and that there is nodifference between forward and futures contracts. (Hint: Use an analogous approach to thatindicated for Problem 13.19.)If the price of currency A expressed in terms of the price of currency B follows the process
dS = (rp —ra)Sdt + oS dz
where r, is the risk-free interest rate in currency A and rg is the risk-free interest rate in currency B.What is the process followed by the price of currency B expressed in terms of currency A?Would you expect the volatility of a stock index to be greater or less than the volatility of atypical stock? Explain your answer.Does the cost of portfolio insurance increase or decrease as the beta of the portfolio increases?Explain your answer.Suppose that a portfolio is worth $60 million and the S&P 500 is at 1200. If the value of theportfolio mirrors the value of the index, what options should be purchased to provide protectionagainst the value of the portfolio falling below $54 million in one year’s time?Consider again the situation in Problem 13.24. Suppose that the portfolio has a beta of 2.0, thatthe risk-free interest rate is 5% per annum, and that the dividend yield on both the portfolio and
the index is 3% per annum. What options should be purchased to provide protection against thevalue of the portfolio falling below $54 million in one year’s time?
Show that the put—call parity relationship for European index options ise+ Ke = p+ Spe?
where q is the dividend yield on the index, ¢ is the price of a European call option, p is the priceof a European put option, and both options have strike price K and maturity 7.Options on Stock Indices, Currencies, and Futures 293
13,27.
13.28.
2BNoSs
13.30.
13.31.
13.32.
13.33,
13.34.
13.35.
13,36,
13,37.
weeooo
13.39.
13.40.
an an oantion on the ven—enro exvchanoe rate bs ded Lian tea ek ek Vee +. Can afi option on the yen—curo exchange rate be created from two options, one on the dotlar—eur
Suppose you buy a put option contract on October gold futures with a strike price of $400 perounce, Each contract is for the delivery of 100 ounces. What happens if you exercise when theOctober futures price is $377 and the most recent settlement price is $380?
Suppose you sell a call option contract on April live-cattle futures with a strike price of 70cents per pound. Each contract is for the delivery of 40,000 pounds. What happens if thecontract is exercised when the futures price is 76 cents and the most recent settlement price is75 cents?
Consider a two-month call futures option with a strike price of 40 when the risk-free interest rateis 10% per annum. The current futures price is 47. What is a lower bound for the value of thefutures option if it is (a) European and (b) American?
Consider a four-month put futures option with a strike price of 50 when the risk-free interest rateis 10% per annum. The current futures price is 47. What is a lower bound for the value of thefutures option if it is (a) European and (b) American?
A futures price is currently 60. It is known that over each of the next two three-month periods itwill either rise by 10% or fall by 10%. The risk-free interest rate is 8% per annum. What is thevalue of a six-month European call option on the futures with a strike price of 60? If the call wereAmerican, would it ever be worth exercising it early?
In Problem 13.31, what is the value of a six-month European put option on futures with a strikeprice of 60? If the put were American, would it ever be worth exercising it early? Verify that thecall prices calculated in Problem 13.3] and the put prices calculated here satisfy put—call parityrelationships.
A futures price is currently