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The Pricing of Foreign Currency OptionsAuthor(s): Angelo Melino and Stuart M. TurnbullSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 24, No. 2(May, 1991), pp. 251-281Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/135623 .

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The pricing of foreign currency options

ANGELO MELINO IUniversity of Toronto S TUART M. TIURNB ULL Queen's University and University of Toronto

Abstract. This study examines the assumption that the exchange rate follows a log-normal probability distribution; and it tests whether different stochastic specifications translate into important differences in implied option prices. We investigate a class of processes, which includes the log-normal probability distribution as a limiting case. None of the models performs particularly well. The main problem appears to be that the volatility estimates from actual exchange rate data are significantly smaller than those implied by observed option prices.

La formation du prix des options sur des devises etrangeres. Cette etude examine le pos- tulat que le taux de change suit une courbe de probabilite log-normale et verifie si des specifications stochastiques differentes se traduisent en des diff6rences importantes dans les prix des options qui en decoulent. Les auteurs examinent une famille de processus dont le processus log-normal est un cas limite. Aucun des modeles ne donne des resultats particu- lierement bons. Le probleme principal semble etre que la volatilite des evaluations a partir des donnees sur les taux de change reel est plus faible et de maniere significative que celle des evaluations derivees des prix d'options observes.

I. INTRODUCTION

Under certain assumptions, foreign currency options can be replicated by a suitably chosen and constantly adjusted portfolio of domestic and foreign bonds. Arbitrage then dictates that the foreign currency option's price is exactly the cost of purchas- ing the replicating portfolio. From this insight a partial differential equation can be derived describing the value of the foreign currency option, for a given stochastic process characterizing movements in the exchange rate.

Financial support for this paper was provided by the Social Sciences and Humanities Research Council of Canada. We are indebted to seminar participants at the Australian Graduate School of Management, Southern Methodist University, Univeristy of Waterloo, and the Canadian Econo- metric Study Group meetings, as well as to an anonymous referee for helpful comments. Ken Vetzal provided excellent research assistance. We are responsible for any remaining errors.

Canadian Joumal of Economics Revue canadienne d'Economique, XXIV, No. 2 May mai 1991. Printed in Canada Imprime au Canada

0008-4085 / 91 / 251-281 $1.50 ? Canadian Economics Association



252 Angelo Melino and Stuart M. Tumbull

Regardless of the stochastic process generating exchange rates, option prices must satisfy certain inequalities or boundary conditions to preclude the opportunity for profitable arbitrage.1 Bodurtha and Courtadon (1986) examine whether foreign currency option prices satisfy these conditions, using transactions data from the Philadelphia Exchange (PHLX) over the period 28 February 1983 to 14 September 1984. Almost all call option prices satisfied the boundary conditions, but they found that a large number of put option prices did not (as much as 8.5 per cent of the sample for some countries). Most of the violations, however, were small and could be accounted for by plausible estimates of transactions costs. The specification of the stochastic process for exchange rates narrows the interval that option prices must satisfy to a specific point forecast. Bodurtha and Courtadon (1987) examine the performance of both the American and the European pricing models, assuming that the exchange rate can be described by a log-normal probability distribution. They found that neither pricing model performed very well. The average ratio of the absolute forecast error to the actual option price for puts and calls was about 20 per cent for both models. In addition, Bodurtha and Courtadon (1987) found a variety of systematic biases related to the spread between the option's exercise price and the spot currency price. They suggest that these biases are most likely due to the fact that rate of change in a foreign currency spot price is not normally distributed.

Boothe and Glassman (1987) provide a recent review of the stylized facts about the empirical distribution of exchange rate changes. They consider a variety of exchange rates and document the distribution of exchange rate differences (expressed in logarithms) over daily, weekly, monthly, and quarterly intervals for the period 1973-84. They report that the distribution of exchange rate differences has much fatter tails than the normal when daily differences are examined, but that the excess kurtosis declines with the longer differencing intervals. Since the monthly difference is simply the sum of the daily differences, the tendency of the kurtosis to decline with the length of the sampling interval suggests that the average of a large number of daily exchange rate changes is approximately normally distributed. This casts doubt on the suggestion of Westerfield (1977), Bagshaw and Humpage (1986), and So (1987) (among others) that stable paretian distributions should be used to model the distribution of exchange rate changes. Boothe and Glassman (1987) find evidence to support the suggestion of Press (1967), McGarland et al. (1982), and Engle (1982) of using finite but heteroscedastic variances to generate the fat tail. Although empirically motivated, the literature on how best to model the exchange rate distribution has important consequences for pricing foreign currency options. If exchange rate movements cannot be described by a log-normal distribution, then this implies that the type of option pricing model used by Bodurtha and Courtadon (1987) is misspecified (Merton 1976; Ball and Torous 1985).

1 As noted in Galai (1978) and Halpern and Turnbull (1985) these tests are not valid for testing market efficiency. They can indicate only whether the option and underlying asset markets are not synchronized or are not continuously in equilibrium.



Pricing of foreign currency options 253

The first objective of this study is to examine the assumption that the exchange rate follows a log-normal probability distribution. We investigate a class of processes considered by Feller (1951), modified to have an absorbing barrier at the origin. These processes have a variance that depends upon the level of exchange rates and are often referred to as 'constant elasticity of variance' (CEV) processes; the log-normal process is a limiting case. Since the level of the exchange rate changes from day to day, this specification will provide heteroscedastic variances that can potentially explain the observed leptokurtosis. Moreover, since the level of the exchange rate is serially correlated, the specification can also generate autoregressive conditional heteroscedasticity. As well shall see below, this is also an important feature of the data, although it has been largely ignored in the empirical literature cited above. Many specifications that have been suggested for the exchange rate distribution, such as independent stable paretian increments or adding a jump component to the log-normal diffusion, can account for the observed leptokurtosis but would not lead to conditional heteroscedasticity. We use daily data on five different exchange rates to estimate the parameters describing the probability distributions. We also estimate the parameters over different periods to see if there is any validity to the concern voiced by some financial institutions that the models suitable for pricing short-term currency options may be inappropriate for pricing long-term currency options, because of parameter instability.

The second objective of this paper is to examine whether the different stochastic specifications translate into important differences in the implied option prices. The differences in using a log-normal model to predict option prices rather than other members of the CEV family will be closely related to the level of the spot exchange rate, which may account for the systematic biases reported by Bodurtha and Courtadon (1987). The estimated parameters of the log-normal and various other processes are used to price the corresponding foreign currency option and the predictions are compared with observed trades. In addition, we compare the performance of the American and European pricing formulas.

The paper is organized as follows. Section II reviews briefly the pricing of foreign currency options. The data are described in section III. The econometric estimation of the stochastic process describing changes in the exchange rate is described in Section iv, and the results of the option pricing model are presented. Section v contains a brief summary and describes the motivation for a companion paper (Melino and Turmbull 1990).

II. THE PRICING OF FOREIGN CURRENCY OPTIONS

In pricing foreign currency options the following assumptions are often made:

Al. No transaction costs, no differential taxes, no borrowing or lending restrictions, and trading takes place continuously.

A2. The term-structure of interest rates in both the domestic and foreign country are flat and non-stochastic.



254 Angelo Melino and Stuart M. Tumbull

A3. The underlying state variable is the spot exchange rate.

A4. The exchange rate S can be described by the stochastic process dS = B' S dt + a S dZ, where B' is the instantaneous mean: a2 is the instantaneous variance; and Zs is a standard Wiener process. It is also assumed that B' and a are constants, implying that the stochastic process describing the evolution of the spot rate over time is log-normal.

Given the above assumptions, the partial differential equation describing the value of a foreign currency option is identical in form to the equation for an option written on a stock that pays a continuous dividend such that the dividend yield is constant. Thus from Merton (1973), the price of a European call option is given by

c(S, T; rF, rD, E) = exp (-rF T) S N(dl) -exp (-rD T)E N(d2) (1)

and for a European put option

p(S, T; rF, rD, E) = -exp (-rF T) S N(-dl) + exp (-rD T)E N(-d2), (2)

where rF(rD) is the foreign (domestic) risk-free interest rate per unit time; T is the maturity of the option: T _ T - t, where T1 is the date the option matures and t the current time; E is the exercise price; d1 _ [ln (S/E) + (rD - rF + 2/2)T]/(uVT);

d2 =_ d, - av'; and N(.) is the cumulative normal distribution function. (See Biger and Hull 1983; Garman and Kohlhagen 1984; Grabbe 1984; or Shastri and Tandon 1986.)

The first objective of this paper is to relax the assumption (A4) about the s- tochastic process describing the evolution of the exchange rate. Marsh and Rosen- feld (1983) have addressed the issue of modelling different types of stochastic processes for interest rates. Drawing on this work, it is assumed that the stochastic process for the exchange rate is of the form

dS - (A'S-'1- + B'S)dt + aSO12dZ, S _ O, (3)

where A', B' and 3 are constants, and 0 ? _ 2. If A' = 0, (3) is identical to the process considered by Emmanuel and MacBeth (1982). For the case: = 2, (3) becomes

dS = (A'+B') S dt+ a S dZ,

implying that S is log-normally distributed; note that in this case it is not possible to estimate A' and B' separately.2

2 If 3 = 1, then (3) is identical to the square-root process considered by Cox and Ross (1976); if 0 = 0, then (3) describes a process that is normally distributed.




Given the form of (3), the partial differential equation describing the market value of the option is as follows (see Emmanuel and MacBeth 1982):

2 U2 S, Css + S (rD -rF)Cs + Ct-rD C = 0 (4)

Note that the above equation does not explicitly depend upon the mean of the process describing changes in the exchange rate. If 03 2, then (4) would be isomorphic to Merton's dividend yield equation (see Merton 1973; Smith 1976). The specification of the stochastic process for the exchange rate affects the partial differential equation for the option via the variance term u2 Sf3. It is possible to solve (4) using numerical methods, given the appropriate boundary conditions.3

A variety of other extensions of assumptions A1--A4 are imaginable, such as explicitly taking into account the effects of stochastic domestic and foreign interest rates, or augmenting the set of state variables required to describe the transition probabilities for the exchange rate. These extensions bring with them a substantial increase in complexity, particulary for the empirically relevant case of American options. It is extremely useful to determine the importance of the log-normality assumption. If predicted option prices can be significantly improved by modest generalizations of assumption A4, there would be little need to consider more demanding extensions.

III. DATA DESCRIPTION

Our empirical work requires the following data: (1) spot exchange rates for Britain, Canada, Germany, Japan, and Switzerland; (2) the term structure of interest rates for the United States and each of the five foreign countries above; and (3) prices of exchange rate options on each of the five foreign currencies.

Daily data on each of the five spot exchange rates for the period 2 January 1979 to 25 June 1986 were obtained from the Bank of Canada. These data are averages of noon bid-ask rates quoted by dealers that were surveyed by the Bank of Canada. We used these data to identify the stochastic processes for the five exchange rates. Note that we did not use these data directly to price the exchange rate options (see below). Table la provides descriptive statistics for each of the five spot exchange rates (S) over various sample periods. Except for the Japanese yen, the data are measured as the number of u.s. cents per unit of the foreign currency. For Japan, the data are expressed as the number of u.s. cents per 100 yen. Three broad features of the data stand out: (1) the average daily change was negative for four of the five exchange rates - the yen being the only currency to appreciate against the u.s. dollar; (2) the average daily change was swamped by the standard deviation of the daily change for all five currencies (in fact, the coefficient of variation on daily changes ranges from about 30-100 over the five countries); and (3) the variance of daily changes is highly unstable over different subsamples.

3 For a description of the application of numerical methods, see Tumbull (1987, chap. 6).



256 Angelo Melino and Stuart M. Turnbull

TABLE la

Exchange rate properties, descriptive statistics

79/01/02-86/06/25 (1887 observations)

Standard Minimum Maximum Country Variable Mean deviation value value

Britain S 174.90 38.37 105.30 245.35 (pound) AS -0.02731 1.20 -7.9781 5.8661

(AS)2 1.4398 3.26 0.00000 63.649

Canada S 80.409 4.74 69.498 87.719 (dollar) AS -0.00637 0.194 -0.92262 1.0732

(AS)2 0.03784 0.078 0.00000 1.1518

Germany S 43.458 7.95 28.969 58.580 (mark) AS -0.00534 0.298 -1.3210 1.6317

(AS)2 0.08881 0.188 0.00000 2.6625

Japan S 44.043 4.47 36.010 62.094 (100 yen) AS 0.00453 0.294 -1.1930 1.5309

(AS)2 0.08678 0.192 0.00000 2.3438

Switzerland S 50.396 7.26 34.204 64.582 (franc) AS -0.00378 0.390 -2.0139 2.0522

(AS)2 0.15187 0.309 0.00000 4.2117

79/01/02- 82/09/27- 83/02/28- 84/02/10- 82/09/27 86/06/25 84/02/10 85/01/24

Country Variable Mean Mean Mean Mean

Britain S 208.43 141.366 149.741 131.439 (pound) AS -0.03574 -0.02041 -0.04641 -0.12615

(AS)2 1.7631 1.1176 0.65367 0.70327

Canada S 84.035 76.785 80.979 76.810 (dollar) AS -0.00338 -0.00953 -0.00504 -0.01972

(AS)2 0.04549 0.03019 0.01027 0.02778

Germany S 49.454 37.458 38.444 34.954 (mark) AS -0.01678 0.00562 -0.02048 -0.02026

(AS)2 0.10342 0.07432 0.03661 0.05848

Japan S 44.352 43.727 42.099 41.897 (100 yen) AS -0.01546 0.02399 0.00036 -0.01396

(AS)2 0.09755 0.07619 0.03267 0.03603

Switzerland S 55.680 45.107 46.886 42.132 (franc) AS -0.01722 0.00933 -0.01876 -0.03058

(AS)2 0.18506 0.11865 0.05753 0.06931

Table lb describes the dynamic properties of the five exchange rates. To keep things manageable, we report only the calculations based on the entire sample. The qualitative features, however, appear to be stable over subsamples. The Box-Ljung (K) statistics give the marginal significance level (i.e., the critical value at which we would just reject the null hypothesis) of a test that the first k autocorrelation-



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s are generated by a white noise process. We report the statistic for twelve and twenty-four autocorrelations. Once again, three broad features of the data stand out: (1) all five spot exchange rates (S) have an autocorrelation function that does not die out quickly which indicates a unit root (i.e., random walk component) in the level process; and (2) all five first differenced exchange rates (AS) are close to white noise. Although some of the marginal significance levels are very small, the estimated serial correlations themselves are just about zero. Finally, (3) the daily variances of the difference exchange rates, approximated by (AS)2, display conditional heteroscedasticity and have a rather complicated serial correlation structure. The hypothesis that (AS)2 is a white noise process can be rejected for all the e- conomies. Note that many suggestions for the exchange rate distribution, such as independent stable paretian increments (Westerfield 1977; Bagshaw and Humpage 1986; So 1987) or adding a jump component to the log-normal diffusion process (Bodurtha and Courtadon 1987; Tucker and Pond 1988), which might account for leptokurtosis would not lead to conditional heteroscedasticity.

To price foreign currency options it is necessary to obtain data on simultaneous option prices and currency spot prices and domestic and foreign interest rates for default-free claims matching the maturities of the option contracts. Our source for the simultaneous option prices and currency prices is the transaction surveillance report compiled by the Philadelphia Exchange and described in Bodurtha (1984). For each option trade the following is recorded (inter alia): the date of trade, currency identification, maturity, exercise price, option price, and the value of the last spot exchange rate quote reported by Telerate from the interbank market. Our sample of option trades runs from 28 February 1983 to 24 January 1985.

Daily closing quotes for u.s. treasury bills with maturities of 90, 180, 270, and 360 days were used to calculate the u.s. default-free rate of interest. For the foreign default-free interest rate, daily settlement prices for foreign futures contracts traded on the IMM were collected. These prices were used to impute a foreign default-free interest rate. This procedure requires two assumptions: first, forward and future prices are identical; second, the interest rate parity theorem holds for these currencies. If interest rates are stochastic, then forward and future prices will, in general, differ.4 However, Cornell and Ranganum (1981) have shown that these differences are small. The validity of the interest rate parity theorem has been extensively examined and has been found to hold reasonably well, especially for the Eurocurrency markets (see Fratianni and Wakeman 1982).

For options with very short maturities, that is, less than two days, the imputed foreign interest rates were often negative, or very large. This could be due to a minor timing problem (options expired on a Friday but the futures contract matured on the subsequent Monday), or, as suggested to us by a number of practitioners, to some peculiarities in the futures market as investors unwind their positions just before maturity. It was decided to drop all observations of options with a maturity of five days or less. Observations were also dropped if there were obvious

4 This follows from the work of Jarrow and Oldfield (1981).



Pricing offoreign currency options 259

TABLE 2

Philadelphia option data, 83/02/28-85/01/24

Minimum No. of No. of Contract contract calls puts Total size price change

British 1,725 840 2,565 12,500 $6.25 options (9,125) (4,560) (13,685)

Canadian 2,450 2,028 4,478 50,000 $5.00 options (2,548) (2,230) (4,778)

German 3,979 1,323 5,302 62,500 $6.25 options (21,788) (7,820) (29,608)

Japanese 2,112 674 2,786 6,250,000 $6.25 options (11,555) (3,700) (15,255)

Swiss 2,295 746 3,041 62,500 $6.25 options (13,073) (4,834) (17,707)

12,561 5,611 18,172 (58,089) (23,144) (81,233)

NOTE: Figures in parenthesis denote the total numbers of options.

recording efrors, such as the option trading after its recorded maturity date. To save computational expense, we did not use all of the remaining available trades, except for the Canadian options. For the other four countries we sampled every fifth data point. The total number of transactions for each option is shown in Table 2.

IV. EMPIRICAL RESULTS

1. Econometric estimation The continuous time model, equation (3), can be estimated using maximum likelihood.5 Consider a transformation y S2- for 0 ,B < 2 so that

d y-(c + b y) dt + a1y d Z, (5)

5 We also estimated two discrete-time models,

ASt = (A1S{-0j?) + B1 StAt)At + -

and

ASt = (A1 + B1 St-At)At + US012 0

where AS, St - St-At; et NID(O, At); and the time difference is set equal to the number of days between observations. Trhe first model is just a discrete time approximation to (3). The second model is very similar, except that it allows for true mean reversion, regardless of the value of f. An advantage of these discrete time models is simplicity: their parameters can be estimated by generalized least squares for a given value of f, using standard regression packages. Although these discrete time models do not incorporate the non-negativity of exchange rates, the estimated values of a and the maximized value of the (quasi) log-likelihood were virtually identically to those obtained from the continuous time model and so are not reported.




where c _ (2 - 3) [A' + (1 - O3)u2/2]; b _ (2 - O)B'; and a -(2 - )2u2. Feller (1951) considers the solution to the forward equation defined by

ap _ 2 (6

aP = a [a y p(Y y t)] - aY [(c + b y) p( y, t)], (6)

where a a, /2 = (2- _3)2u2/2, and p(y, t) is the probability density function and shows that the solution is given by

p(y(t) = Y I Y(O) = Yo) = (y/A)k/2 exp [-(y + A)/B]Ik [2(yA/B)/2, (7) B

where A yo exp (b t); and B- a[exp (b t) - 1]/b. For k _ 1 - c/a, Ik(-) is the modified Bessel function of the first kind of order k. We require c < a and k ? 0. Given the restrictions on the parameters, it is assumed that there is an absorbing barrier at the origin.

In (3) there are four parameters to estimate: A', B', u, and /. Following Marsh and Rosenfeld (1983), we use maximum likelihood to estimate A', B', and u for a grid of values of 3.6 Using a grid (rather than estimating 3 directly7 ) provides useful information about the sensitivity of the likelihood surface to different choices of /. The grid also facilitates our second objective of examining the sensitivity of the predicted option prices to the choice of the exchange rate distribution.

2. Estimates of the exchange rate process The continuous time maximum likelihood estimates are presented in tables 3 and 4. The likelihood function is easily amended for irregularly spaced observations; so all available data over each of the subsamples were used in the estimation. For / = 2, we report the estimates of the usual log-normal process; that is, dS =

(A'+ B' * S)dt + oSdZ.8 Over the one-year intervals examined in table 3, there is very little evidence to

distinguish the different models; all do equally well, in the sense that the maximized likelihood function is flat over the range of beta. The drift term, B', is poorly estimated, being statistically insignificant from zero in most cases.9 The volatility

6 Marsh and Rosenfeld (1983) report an incorrect density function. In Feller (1951), the term (x/A)-k/2 is misprinted as (4b2x/A k/2. That (7) is a solution to (6) can be checked by direct substitution. As a quick check, the solution for c 0_ has been derived by many authors - for example, see Cox and Miller (1968, 236).

7 Gibbons and Jacklin (1989) show that estimation of 3 is greatly facilitated by a suitable transformation.

8 We also estimated the model at / = 1.99. The values of a and the log-likelihood were virtually identical to those reported for the log-normal case. As a further computational check, we estimated the normal model, that is, dS = (Al + B1 * S)dt + adZ. Although the normal model does not respect the non-negativity of exchange rates, it provided values of a and the log-likelihood that were virtually identical to the / = 0 case.

9 The correlation between A' and B' is close to -1 in every case. However, there is very little correlation between these parameters and a. The null hypothesis of a unit root in the exchange rate processes invalidates the usual standard errors for A' and B' but would not affect our conclusion that they are insignificantly different from zero. Because of block diagonality of the information matrix, taking account of the unit root would not affect inferences about a.



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TABLE 5

Value of beta

Period

79/01/02- 79/01/02- 82/09/27- Country 86/06/25 82/09/27 86/06/25

Britain 0.5* 0* 0* Canada 0* 0* 0* Germany 0.5* 0* 1* Japan 2 0* 2 Switzerland 1* 0* 1.5

* Statistical different from ,3 = 2 at the 99 per cent level

parameters are estimated fairly accurately. For the two short subperiods reported in table 3, the estimated volatility parameter is fairly stable for Britain, Japan, and Switzerland. 10

For the period 2 January 1979 to 25 June 1986 a different picture emerges. Except for Japan, the log-normal model can be rejected in each case.11 Canada appears to be best fit by the : = 0 model; Britain and Germany by the / = 0.5 model; and Switzerland by the B = 1.0 (square root) model. For the first subperiod, 2 January 1979 to 27 September 1982, the log-normal model can be rejected for all countries; this is not the case for the second subperiod, 27 September 27 1982 to 25 June 1986, where the log-normal model cannot be rejected for Japan and Switzerland. See table 5. Again the drift term, B', tends to be poorly estimated, the volatility parameter accurately estimated, and the hypothesis that the parameters were stable over the whole period can be rejected.

The volatility parameters for the log-normal case are somewhat lower than those reported by Bodurtha and Courtadon (1987), who estimated a implicitly from observed option prices for the period 28 February 1983 to 21 September 1983. Converting their estimates from yearly to daily implied standard deviations, using a 365-day calendar year, yields values that are 16 per cent larger for Britain, 23 per cent larger for Switzerland, 35 per cent larger for Germany, 62 per cent larger for Japan, and 85 per cent larger for Canada than the estimates we report in table 3 for a similar sample period. Given the precision of our reported estimates, such a large difference between the estimated and implied volatilities suggests evidence of misspecification.

10 The test statistic is 2(L1 + L2 -LT) X2, where L1 (L2) is the log-likelihood function over the first (second) period: LT is the log-likelihood function over the whole period; and q is the number of parameters estimated q = 3.

11 The test statistic used is 2(L - LC) - xI, where L is the unconstrained log-likelihood function; and LC is the constrained (/3 = 2) log-likelihood function. This is approximate because /3 is not allowed to be greater than two, and we considered only values of /3 in our grid.



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3. Comparison of predicted and actual foreign currency options The options were priced with the continuous time estimates of the volatility parameter given in table 3 for the two subperiods, given the presence of non-stationarity. Descriptive stastistics for the predicted and actual option prices are reported in table 6. The standard deviations of the predicted and actual prices are similar. However, all models underprice the options, although (not surprisingly) the American pricing models do somewhat better than the European ones.

To examine in more detail the differences between observed option prices (C) and predicted option prices (C) two cross-sectional regressions were estimated:

Test 1: C = ao + a, C +

and

Test 2: C-C - 'O + 'l(S-X)/X + Y2(T/IOO) + Y3rD + 1y4rf + e,

where S denotes the spot exchange rate; X the exercise price; T the maturity of the option (measured in days); rD(rF) the domestic (foreign) rate of interest; e are disturbance terms. In test 1, a model provides an unbiased estimate of the actual premium if ao 0 0 and a, = 1. Test 2 examines whether the prediction error is systematically related to the inputs of the pricing model. The interpretation of "yr depends upon whether the option is a call or a put. For call options, a positive value of "y, means that the model under (over) predicts the price of an option that is in (out of) the money. If 1Y2 is positive, then the longer the maturity of the option, the greater the degree of underpricing by the model. If 'Y3('Y4) is positive, the degree of underpricing is an increasing function of the domestic (foreign) interest rate.

The results of tests 1 and 2 are reported in tables 7, and 8, respectively, for the period 28 February 1983 to 24 January 1985. These tests were also performed over the two subperiods, but, given the similarity of the findings, these results are not reported. Because of substantial heteroscedasticity and serial correlation, all standard errors and hypothesis tests reported are based on Newey and West's (1987) covariance matrix estimator. Tests 1 and 2 used only the option trades that satisfied the boundary conditions described in Courtadon and Bodurtha (1986); the results were virtually identical to those when this restriction was not imposed.

Test 1 A model provides an unbiased estimate of the actual premium if ao 0 0 and a, = 1. This hypothesis is tested using the Wald statistic, which is asymptotically distributed as X2(2). Table 7 shows that for almost all cases the null hypothesis can be overwhelmingly rejected, the exceptions being British and Swiss puts. The slope coefficients tend to be close to 1, but because the actual option prices are usually larger than the predicted prices, there is usually overwhelming evidence that the intercepts in test 1 are positive. There does not appear to be any relationship between the rankings of the Wald statistic for different values of 3, and the rankings of the likelihood reported in table 3. Bodurtha and Courtadon (1987) found that for the log-normal distribution, a European option pricing model tended to underprice



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options and an American option pricing model tended to overprice options. No such reversal is found in table 7.

Test 2 A number of general comments can be made about the test 2 results reported in table 8. For all countries except Canada, and for calls and puts, the intercept YO is positive and tends to increase as beta increases. For Canada the intercept is negative. The ^Yi coefficient for calls and puts is positive, except for German and Japanese call options, and is often statistically significant. For the European pricing model Yi

is negative for German, Japanese, and Swiss put options. For all countries, and for calls and puts, the 1Y2 coefficient is positive and statistically significant, implying that the longer the maturity of the option the greater the degree of underpricing. It is also observed that 1Y2 is a non-decreasing function of beta. The 1Y3 coefficient for the domestic interest rate is negative for call and put options for all countries except for Canadian put options. For the foreign interest rate, ̂ y4 is positive for call options and negative for put options and statistically significant for all countries, except Canada. For all cases, the Wald test statistic, which is asymptotically chi- square distributed, x2(5), indicates that the joint hypothesis that the coefficients YO,

1Y1, 1Y2, 1Y3, and 1y4 are zero can be rejected.

Non-linearity The option value is in general a non-linear function of maturity, the ratio of the stock price to exercise price, and the domestic and foreign interest rates. A limitation of test 2 is the linear form of the explanatory variables. The best way to model the non-linear form of these variables is not clear. One approach would be to redefine the dependent variable in test 2 to be the proportional error, (C - C)/C, and to estimate the coefficients. This has been done, though the results are not reported in detail because of space limitations. In general, these results showed greater differences across countries and across contracts.

To examine the relationship between the mispricing of the option and the degree of being in or out of the money, a graph of the exchange rate minus the exercise price, (S - X) was plotted against the pricing error, observed option price minus predicted (C - C), for each country and for each type of contract. The results for Britain are shown in figures 1 and 2.12 The X-axis, measuring S - X, was divided into a number of intervals, the size of the interval being chosen in an attempt to equalize the number of observations. In a given interval the mean option pricing error was determined and the mean was plotted on the graph. While it is very difficult to provide a generalization of the graphs for all countries, three comments can be made. First, there does appear to be a non-linear relationship, though the form of the relationship varies over country and type of contract. Second, the American option pricing models outperform the European pricing model, especially for put options. Third, it is difficult to discern any general dependence upon beta.

12 The values for / = 0.5, 1, 1.5 models are not shown since they were bounded between the , 0 and /3 2 models.



274 Angelo Melino and Stuart M. Turmbull

.70

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.................. European

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For put options and in the money call options, except for Japanese call options, the a3 0 model does a slightly better job of describing option prices.

A similar procedure is used to examine the effect of maturity upon the degree of mispricing. The results for Britain, which displayed the most complicated structure of all the countries, are shown in figures 3 and 4. Two general comments can be made. First there is a general positive relationship between maturity and the degree of mispricing. Second, the degree of mispricing seems, in general, to undergo a substantial increase for long maturity options.

Check on interest data In calculating the option prices, the foreign interest rates were determined assuming the validity of the interest rate parity theorem. In a small number of cases the implied interest rate was unusually large. To check that this did not unduly bias the results, tests 1 and 2 were repeated using only observations where the difference between the domestic and foreign interest rate was less than 10 per cent. Similar results to those reported in tables 7 and 8 were obtained and consequently are not reported.

For Canada good interest rate data similar to those used for the United States were available and the options were priced using these data. The results for test 1 are very similar. For test 2 some differences occurred, especially for put options. For call options there was only one substantial change: -Yi remained positive, in- creased in size, and became statistically significant. For put options, the intercept 'Yo switched from negative to positive. The interest rate coefficients also switched sign, with 'Y3 becoming statistically significant but 4 remaining statistically insignifican- t. The sign and statistical significance of the in/out of the money coefficient, '1, and the maturity coefficient, 1Y2, remain unchanged. The null hypothesis that the coefficients were zero was overwhelmingly rejected.

IV. CONCLUSIONS

The key parameter required to price foreign currency options is the instantaneous variance of exchange rate changes. There is ample heuristic evidence to suggest that this variance changes' over time. The usual log-normal representation posits that the instantaneous variance is proportional to the square of the level of the exchange rate. This paper has considered a family of processes where shifts in the instantaneous variance are also due entirely to changes in exchange rate levels, but the elasticity of the response is allowed to be less than in the log-normal case. The evidence suggests that this is indeed the case, at least when we consider long enough samples, and the log-normal model for exchange rates is rejected almost uniformly. This may have important implications for financial institutions offering long-term (say, five-seven-year) foreign currency options.

The transition probabilities generated by the different models are very similar for the relatively short horizons for which foreign currency options are currently traded (i.e., less than nine months), so all these models generate very similar predictions for option prices. None of the models performs particularly well, but the




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American models appear to do somewhat better than the European ones. The main problem appears to be that the volatility estimates from actual exchange rate data are significantly smaller than those implied by observed option prices.

A natural direction for future research is to investigate the consequences of treat- ing the volatility parameter itself as a diffusion process. We do so in a companion paper (Melino and Turnbull 1990).

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Garman, M.B. and S.W. Kohlhagen (1983) 'Foreign currency option values.' Journal of International Money and Finance 2, 231-38

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Jarrow, R.A. and G.S. Oldfield (1981) 'Forward contracts and futures contracts.' Journal of Financial Economics 9, 373-82




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Melino, Angelo and Stuart M. Turnbull (1990) 'The pricing of foreign currency options with stochastic volatility.' Journal of Econometrics 45, 239-65

Merton, Robert C. (1973) 'Theory of rational option pricing.' Bell Journal of Economics and Management Science 4, 141-83 (1976) 'The impact on option pricing of specification errors in the underlying stock price returns.' Journal of Finance 31, 125-44

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Shastri, Kuldeep and Kishore Tandon (1986) 'Valuation of foreign currency options: some empirical tests.' Journal of Financial and Quantitative Analysis 21, 145-60

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