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The Valuation of Currency OptionsNahum iger and John Hull

Both Nahum Biger and John Hull are Associate Professors of Financein the Faculty of Administrative Studies, York University, Canada.

IntroductionSince Black and Scholes [1] published their path-breaking paper, option pricing theory has receivedconsiderable attention in the literature. Many authorshave shown how the basic Black-Scholes model can beextended with its underlying assumptions being re-laxed. The model has also found many applications infinance. Smith [6] provides a good overall review ofthe subject.Options on a foreign currency can be defined in thesame way as options on a stock. For example, a Euro-pean call option on a foreign currency is an option tobuy one unit of the currency on a predeterm ined date ata predetermined exchange rate. At the time of writingthere is no well-organized ma rket for foreign currencyoptions. However, the Philadelphia, Montreal andVancouver stock exchanges have all submitted propos-als for the creation of such a market.The prices of foreign currency op tions can be im por-tant in determining the values of other financial con-tracts. Feiger and Jacquillat [2] consider on e such con-tract, the currency option bond. (This is a bond wherethe holder can choose the currency in which couponsand principal are paid according to a pre-determinedexchange rate.) Feiger and Jacquillat point out that atwo-currency, currency option bond is equivalent to a

single-currency bond plus a foreign currency optionThus:P = B -I- cp

where P is the price of a bond paying either 1 or p time T, B is the price ofapure discount bon d payingat time T and c is the price of a European call option topurchase 1 for a dollar price of '/p at time T.Options would add completeness to the foreign ex-change market. For example, by combining a long(short) position in a currency with a put (call) option , a

corporation or private investor could limit downsiderisk while benefiting from favorable exchange move-ments. This partial hedge would be an alternative tothe total hedge that can be obtained using forwardmarkets. F eiger and Jacquillat [2] point out that foreigncurrency options might also be attractive to a corpora-tion that is uncertain whether it will have a long posi-tion in a currency e.g., bec ause it is biddin g for foreign currency denominated contract). They showthat the combination of a forward contract to sell thecurrency at time T and a call option to buy it at time Tprovides a hedge not available with forward contractsalone.

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BIGER, HULL/VALUATION OF CURRENCY OPTIONS

Foreign currency options have received relativelylittle attention in the literature. Feiger and Jacquillat12] develop a valuation model for two-currency, cur-rency option bond s, assuming a joint stochastic p roc-ess for the exchange rate, home interest rate and for-eign interest rate. More recently Stulz [9] hasdeveloped a series of analytical formulas for Europeanput and call options on the minimum or maximum oftwo risky assets. These can be applied to value curren-cy options.

This paper first provides a direct derivation of valu-ation formulas for European put and call foreign ex-change options using the Black-Scholes methodology.It then shows that the same formulas can be derived byassuming the expectations theory of exchange ratesand the capital asset pricing model. The formulas areillustrated with an example. The paper assumes a sto-chastic process for just one v ariable. This produc es farsimpler valuation formulas than those of Feiger andJacquillat |2]. The approach also has the advantagethat it provides useful insights into the key role playedby the forward exchange rate in the valuation process.Valuation Using Black-ScholesMethodology

In this section we value European put and call op-tions on a foreign currency under the following as-sumptions:

a) the price of one unit of foreign currency followsa Geometric Brownian Motion;b) the foreign exchange market operates continu-ously with no transaction costs or taxes;c) the risk-free interest rates in both the foreigncountry and the home country are constant during thelife of the option.In the Appendix we show how the assumptions in (c)

can be relaxed.When valuing options on a stock. Black and Scholes|1] make the following assumptions: (i) the stock fol-lows a Geometric Brownian Motion, (ii) there are nopenalties for short sales, (iii) transaction costs and tax-es are zero, (iv) the market operates continuously, (v)the risk-free interest rate is constant, and (vi) the stockpays no dividend. They show that it is possible tocreate a riskless hedge using only the stock and Euro-pean call options written on the stock. The holdings inthe hedge must be continuously adjusted so that theratio of the number of stock held to call options sold isalwaysddd where S is the stock price and c is the callprice. In equilibrium the return on the hedge must bethe risk-free interest rate and this leads to a differential

equation relating c and S. The solution to the differen-tial equation is the well-known Black-Scholes optionpricing formula:

c = S-N ln(S /X ) -H [ra / Tln(S/X) + [r -a / T (1 )

where X is the exercise price , T is the exercise date , a^is the instantaneous variance of the stock's return, r isthe risk-free interest rate and N is the cumulative stan-dard normal distribution function.As Black and Scholes assume that no dividends are

paid on the stock during the life of the option, theirmodel cannot be directly applied to va lue an option ona foreign currency. This is because an investor whowishes to hold a foreign currency should alwayschoose short-term risk-free foreign currency bonds inpreference to holding the foreign currency in somenon-interest-bearing account. A holding of a foreigncurrency can, therefore, be considered as giving a re-turn equal to the foreign risk-free rate and valuing anoption on a foreign currency is, therefore, essentiallythe same problem as valuing an option on a stockpaying a continuous dividend. Merton [5] and Smith[6] consider the latter problem on the assumption thatthe dividend yield, 8, is constant. They show that ariskless hedge can be constructed as above, with thesame hedge ratio, and that the valuation formula be-comes:

c = . . , J ln(S/X) -h [r - 8 + (CTV2)]T| 7 1- e-^^x.N / 1 (S/X) -h [r - 8 - (CTV2)]T1

1 o r J 2 )If the risk-free interest rate that can be earned on the

foreign currency holding, r*, is assumed to be con-stant, the divid end yie ld from an investment in theforeign currency (when m easured in terms of the homecurrency) is constant and equal to r*. Hence, if weredefine variables as follows:S: spot price of one unit of the foreign currency;CT^: instantaneous variance of the return on a for-

eign currency holding;X, T: exercise price and date of a European call op-tion to purchase one unit of thp foreign curren-cy ;r: risk-free rate of interest in the home country;

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36 FINANCIAL MANAGEMENT SPRING 983

then, under the assumptions given at the beginning ofthis section. Equation (2) provides a valuation formulafor a European call option written on the foreign cur-rency when 8 = r*:

ln(S/X) - [r - r*ln(S/X) -I- [r - r* - CTV2 ) ]T1 T T 3

Define F as the forward rate on the foreign currencyfor a contract with delivery date T. Interest rate paritytheory implies:

ln(F/S) = (r - r*)T 4and substituting (4) into (3) the valuation formula re-duces to:

ln(F/X) a / T

ln(F/X) - ((a / T

To value a European put option with exercise priceX and exercise date T, an argument similar to that inMerton [5] relating put and call prices for a stock canbe used. The terminal value of the put option is thesame as the terminal value of a portfolio consisting of:

i) a call option with exercise price X and exercisedate T;

ii) (X F)e '^ of risk-free bonds;iii) a forward contract with delivery date T to sell 1

unit of the foreign currency for price F.Since the value of the forward contract is zero, it fol-lows that the value of the put option, p, is given by:

p = c -F (X - F)e- \ (6)Since 1 - N(q) = N( - q) we obtain:

ln(X/F) -Ip =CT/T

ln(X/F) -a / T

7

It is clear from (5) and (7) that the forward rate playsa central role in the valuation of foreign currency op-tions,c and p depend on F, X,CT,T and r rather than onS, X, a, T and r (which are the parameters involved invaluing stock options). This suggests an insightful ex-tension of the Black-Scholes approach where an inves-tor forms a riskless hedge by combining forward con-tracts with a short position in call options. The detailsare presented in the Appendix. Formulas similar to (5)and (7) are produced without the assumption of theforeign risk-free rate being constant.Alternative Approach

It is interesting that the formulas in (5) and (7) canalso he derived under the following assumptions:a) the covariance between the foreign exchange

rate and the returns from an international marketportfolio is zero;'

h) the spot rate follows a Geometric Brownian Mo-tion with instantaneous standard deviation, a;

c) the international Sharpe-Lintner capital assetpricing model holds.^

Assumption (a) implies that the covariance of thereturns from an option on the foreign currency and thereturns from the international market portfolio is zero.From assumption (c) it follows that options on theforeign currency should be valued at their expectedterminal value, discounted at the risk-free rate, i.e..

c =and

p = O

X) f(S, I So) dS,

So)whereS .is the price of the foreign currency at time T,So is the price of the foreign currency at time O and f isthe probability distribution of Sj conditional on So.

As is well-known, assumption (b) implies thatlnCS^^/So) is normally distributed with standard devi-ation a/ T . The lognormality property of GeometricBrownian Motion has been found appealing in the caseof stocks by many researchers. In the case of a foreigncurrency it is worth noting that it has an added appeal.If the price of the foreign currency expressed in termsof the home currency (= S ) is lognormal then theThisis atbest only approximately true.Recent theoreticalandempiri-cal work suggests thatthecovariancemay benon-zero.See forexampleHansenandHodrick[4] andStuiz18].^See,forexample,Graueret al. [3] for adiscussionof the internationalcapital asset pricing model.

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BIGER HUL L/VALUA TION OF CURRENCY OPT IONS 27

price of the home currency expressed in terms of theforeign currency (= l/S . is also lognormal.Using integrals ofthelognormal distribution that arederived in an appendix to Sprenkle [7] and reproducedby Smith |6] it follows that:c = ln(S/X) + [pa / T

ln(S/X) + [p -a / T

(8)

where p is the expected average growth in the price ofthe foreign currency.From assumptions (a) and (c) it follows that theforward rate, F, is an unbiased predictor of the spotrate at time T. Hence,

pT = In F/S (9)When (9) is substituted into (8) we obtain precisely thesame formula for c as (5). A similar analysis leads tothe same formula for p as (7).ExampleTo illustrate these results we consider a U.S. com-pany that is due to receive Can. $1M in 3 months'time. We assume that the 3-month forward rate for theCanadian dollar is $0.80, that the 3-month risk-freeinterest rate is 2. 5% , and that the standard deviation oflnCS^/So) is 0.02 where, when S is the spot rate at acertain time, S^. is the spot rate 3 months later.The price, p, ofaput option to sell Can. 1for $0.80in 3 months is given by substituting X = 0.80, F =10.80, 1.025-and ujT = 0.02 in equation (7):

P = 1 X 0.80 X N(O.Ol) - 11.025 1.025X 0.80 X N( -O.O l) = 0.0062

Similarly, the price of a put option to sell Can. $1 for$0.79 in 3 months is $0.0025 and the price of a putoption to sell Can. $1 for $0.81 in 3 months is$0.0123.Thus, given an efficiently-functioning market forcurrency options, the company could choose betweena number of different hedging strategies. A t a cost of$6,200 it could ensure that it would obtain a m n mumof $0.80M for the Can. $1M; at a cost of $2,500 it

could ensure that it would obtain a minimum of$0.79M; at a cost of $12,300 it could ensure that itwould obtain a minimum of $0.8lM. All of thesestrategies would be alternatives to using the forwardmarkets whe re, at virtually no cost, the company couldensure that it would obtain exactly$0.80M.Call options could be used in a similar way by acompany due to pay out a certain sum of money de-nominated in a foreign currency at a certain time in thefuture.Summary

The results in this paper provide an elegant app lica-tion of the Black-Scholes methodology. It is interest-ing that the forward rate plays a central role in thevaluation formulas. The generality of the analysis inthe Appendix suggests that the forward rate is centralto the valuation of European put and call options onany income-producing security.References

1. F. Black and M. Scholes, Th e Pricing of Options andCorporate Liabilities, Journal of Political Economy(1973), pp. 637-659.2. G. Feiger and B. Jacquillat, Currency Option Bonds,Puts, and Calls on Spot Exchange and the Hedging ofContingent Foreign E arnin gs, Journal of Finance(1979),pp . 1129-1139.3. F. L. Grauer, R. H. Litzenberger and R. E. Stehle, Sha r-ing Rules and Equilibrium in an International Ca pital Mar-ket Under Uncertainty, Journal of Financial Economics(1976), pp. 233-256.

4. L. Hansen and R. Hodrick, Forw ard Exchange Rates asOptimal Predictors of Future Spot Rates: An EconometricAnalysis, Journal of Political Econo my (1980), pp. 8 2 9 -853.

5. R. C. Merton, Theo ry of Rational Option Pric ing, BellJournal of Economics and Management Science (1973),pp . 141-183.6. C. W. Smith Jr., Option Pricing: A Re view , JournalofFinancial Economics (1976), pp. 3-51.7. C. M. Sprenkle, W arra nt Prices as Indicators of Expecta-tions and Prefere nces, in P. Cootrev, ed.. The RandomCharacter of Stock M arket Prices, Cambridge, Mass, MITPress (1964), pp. 412^74.

8. R. M. Stulz, A Model of International Asset Pric ing ,Journal of Financial Economics (1981), pp. 383-406.9. R. M. Stulz, Option s on the Minimum or the Maximumof Two Risky Assets: Analysis and App lica tion s, yoMr/i/of Financial Economics (\9%2), pp. 161-185.

10. S. M.T um bull, A Note on the Pricing of Foreign Curren-cy Op tions , Working Paper, Department of Econom ics,University of Toronto, (March 1983).

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AppendixAn investor can form a riskless hedge by c omb ining

a long position in forward contracts with a short posi-tion in call options. At any given time, t, he mustadjust his portfolio so that the ratio of forward con-tracts held to call options sold is ^d^ where F, =Pg-r(T - gj^jj p jg {j^g forward rate at time t for acontract with delivery date T. (This hedge ratio isexplained by the fact that when F increases by 8F thevalue of a forward contract increases by 8Fe''

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