Misspecification and the pricing and hedging of long-term foreign currency options

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Journal of International Money and Finance, Vol. 14, No. 3, pp. 373-393, 1995

I N E M A N N

Elsevier Science Ltd Printed in Great Britain. 0261-5606/95 $10.00 + 0.00

0261-5606(95)00003-8

Misspecification and the pricing and hedging of long-term foreign currency options ANGELO MELINO

Department of Economics and Institute for PolicyAnalysis, University of Toronto, Toronto M5S 1A1, Canada AND STUART

M

TURNBULL*

School of Business, Queen's University, Kingston K7L 3N6, Canada This study examines the effects of stochastic volatility upon the pricing and hedging of long-term foreign currency options. The traditional method of pricing such options uses a constant volatility option model with an implicit volatility derived from a short-term traded option. We show that the traditional method leads to small pricing errors for short- and medium-term options, but does a poor job in pricing long-term options. The traditional method also does a poor job for all maturities in determining the derivatives of the option's value with respect to the exchange rate (delta) and the volatility (vega). The errors in calculating these derivatives, which are used in forming the replicating portfolio, lead to large and costly errors in error hedging. (JEL 612).

Financial institutions offer an increasing array of long-term contingent contracts to clients. Long-term contingent contracts now exist for a variety of different assets: interest rate caps, index derivatives, commodity contracts and foreign currency contracts. W h e n financial institutions offer such contracts, two questions arise: how to offset the risk and what price to charge? According to received theory, the answers to these two questions are inextricably linked. Suppose it is possible to construct a replicating portfolio, that is, a portfolio that matches exactly the payoff to the option in all states of the world. Financial institutions can then act basically as a broker - - write the option for clients and simultaneously purchase the replicating portfolio. Since the institution's position is perfectly hedged, the price of the option should be exactly the cost of the replicating portfolio. If the option is actively traded, the * We would like to thank the seminar participants at the Australian Graduate School of Management, Toronto, Wharton and Baruch for their comments. This research was supported by SSHRC and by the Financial Research Foundation of Canada.

Pricing and hedging of long-term foreign currency options: A Melino and S M Turnbull

replicating portfolio might consist of simply purchasing the option in the market. Whether or not it is actively traded, financial institutions may be able to use their specialized knowledge and comparative advantage in the technology of trading to manufacture a replicating portfolio from other assets which are actively traded. Option theory shows how to construct such a replicating portfolio by implementing a dynamic trading strategy. While option theory is well developed, it requires that a variety of choices be made. The challenge in picking a reliable price and hedging strategy arises due to difficulties (i) in choosing the state variables that govern the evolution of the option's value; (ii) in choosing the specification of the transition probabilities for the state variables; (iii) in determining the risk premia for those state variables that are not traded assets; and (iv) in estimating the values of unknown parameters and unobserved state variables. Practitioners face a good deal of freedom and not much guidance in these choices. Typically, they use a variant of the Black-Scholes model, implying that (i) there is only one state variable; (ii) it follows a geometric Brownian motion; (iii) there are no risk premia arising because of non-traded state variables; and (iv) the only unknown parameter is the variance of the instantaneous rate of return. The variance parameter is usually estimated implicitily from the observed prices of traded options. These options typically have maturities measured in terms of weeks or months. The implicit variance are then used to price long-term options. Many financial institutions have expressed concern about a strategy that uses 'short-term' implicit variances to price long-term options. Are these concerns justified? How can we assess the hedging and pricing performance of using this strategy? A direct test of the use of the constant volatility model with implied variances (or any other pricing strategy) for pricing long-term options would compare predicted option prices with prices observed in the market. At this time, such a test is not feasible since there is no active secondary market for long-term options. We provide a procedure for comparing different models that does not require data on traded option prices. In this paper we look at the pricing and hedging of long-term foreign currency options. This is a large and growing market. We focus on the traditional 'Black-Scholes' or constant volatility model and a competitor based on the assumption that volatility is stochastic. Of course, there are many other competing models. We cannot consider them all in this paper. However, the methodology that we describe is very general. It can be applied to evaluate any number of models for all forms of long-term options which are not actively traded. Before describing our procedure, and to motivate our particular focus, it is useful to note that indirect evidence about the constant volatility model is available in two forms: tests based on short-term traded options; and, tests of the assumed geometric Brownian motion specification for exchange rates. This evidence casts doubt on the adequacy of the traditional option pricing model and motivates an alternative based on the assumption that volatility is stochastic.

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Bodurtha and Courtadon (1987) assume that exchange rates behave like geometric Brownian motions. They examine the performance of the corresponding American and European foreign currency option pricing models, using traded options and implicit variances based on the previous day's market prices. Their findings are mixed. Although the average bias in the prediction is small, the dispersion of the prediction errors is substantial. The average of the absolute forecast error to the actual price for puts and calls was about 20 percent for both models. While disappointing, the performance of these models using implicit volatilities is markedly superior to that obtained using historical volatilities - - see Melino and Turnbull (1991). The available evidence overwhelmingly rejects the hypothesis that exchange rates follow geometric Brownian motion. 1 In daily data, the percentage changes in exchange rates are approximately uncorrelated, but they display conditional heteroscedasticity and much fatter tails than the normal distribution. Melino and Turnbull (1990) show that these statistical regularities are captured by a stochastic volatility model. They also find, using transactions data, that the stochastic volatility option model can explain observed option prices with less error than a traditional option pricing model with historical estimates of volatility.2 Both the constant volatility model with implicit variances and the stochastic volatility model can be used to price and hedge long-term options. Without data on traded long-term foreign currency option prices, how can we choose between them? Our approach draws upon the link between pricing an option and replicating it. Rather than focus on pricing errors, we look at the accuracy of hedging strategies. Our test of a candidate model is based on the divergence between the paths of the predicted option price and the value of the model's replicating portfolio. If the model is specified correctly, a position in the option can be perfectly hedged by an offsetting position in the replicating portfolio. With misspecification, the supposedly hedged portfolio will in fact generate random cash flows over the life of the option. The size of these cash flows is an economically interesting way of measuring the importance of misspecification. Our evidence on model performance is divided into two parts. We first provide some simulation evidence. How well does the constant volatility model with implicit variance estimates perform in the presence of misspecification? We generate exchange rate data assuming that volatility is stochastic and examine the pricing and hedging errors made by the constant volatility model. The implicit variance is re-estimated each period using the true price of a short-term at-the-money option. These simulations help answer the question raised by financial institutions about the use of 'short-term' implicit volatility to price long-term options. We find that the use of implicit volatility, when volatility is stochastic, leads to good predictions for the prices of short-term options, but not for long-term options. Although using implicit volatilities sometimes tempers the effects of misspecification on pricing, they are not a panacea. Misspecification leads to substantial hedging errors, even for those cases where the use of implicit volatilities provides accurate prices. In our

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Pricing and hedging of long-term foreign currency options: A Melino and S M Turnbull

simulations, the hedging error is large and increases with the maturity of the option. It should come as no surprise that models can lead to accurate prices, yet provide poor guidance on hedging. Consider an extreme example. With active markets, any procedure that amounts to quoting traded prices will give the right price. But, knowing the true price alone provides no guidance on how to construct a replicating portfolio. Simulations provide a useful controlled environment in which to illustrate what we can hope to learn from using our procedure. However, the hedging performance based on historical experience is perhaps of more interest. The second part of the examination replaces the simulation data with actual exchange rate data. We use the data that underlie the parameter estimates employed in our simulations, and reach the same conclusions. Based on historical data, the stochastic volatility model leads to accurate hedging. The constant volatility model does not. Section I gives a brief review of the pricing of foreign currency options. The general methodology for our tests based on hedging performance is set out in Section II. The simulation results are presented in Section III and the results using real data in Section IV. In Section V, we discuss some of the practical implications and weaknesses of our results. In particular we discuss the question of how to price long-term options when we know the option model is misspecified. While the discussion centers around foreign currency contracts, the issues are pertinent for all forms of long-term derivatives. Section VI concludes.

I. The pricing of foreign currency options We assume the following: A1. A2. A3. A4.

No transactions costs, no differential taxes, no restrictions, and trading takes place continuously. The term structure of interest rates in both the country are flat and non-stochastic. The state variables are the spot exchange rate volatility (v). The transition probabilities for the state variables ( 1}

borrowing or lending domestic and foreign (S), and the level of are summarized by

dS = (a + bS) dt + vS d Z s ,

and (2)

dv = ~ ( v ) d t + ~r(v)dZo,

where Z s and Z v are standard Wiener processes whose increments have instantaneous correlation p; a and b are parameters; p~(v) is the instantaneous mean and ~rZ(v) is the instantaneous variance of the volatility process. We require that zero is an absorbing barrier for S. 3 Given A1 to A4, the price of options written on spot foreign currency can be developed using familiar methods. The option price (C) satisfies the following 376

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Pricing and hedging of long-term foreign currency options: A Melino and S M Tumbull

partial differential equation (3)

2 ( v 2 S g C s s + 2 p ° ' v S C s ~ + °'2C"") q- ( r D -- r F ) S C s + (IX(V) --

A~r(v))C~

-

r D C -t- C t = 0

where ro and rF denote the domestic and foreign risk free rates respectively, and A is a risk premium that arises because volatility is not a traded asset. Note that the drift in the exchange rate does not appear in equation (3). The derivation of the option price does assume knowledge of all the parameters of the volatility process (including its drift), along with the level of volatility and the risk premium at each point in time. Note also that our assumption of a constant risk premium does not introduce any arbitrage opportunities; it follows, for example, from ( 1 ) - ( 2 ) and log preferences. To proceed, we must be more specific about the volatility process. We follow Melino and Turnbull (1990) and assume that the volatility process follows (4)

d l n v = ( a + 61nv)dt + y dZ~,

so that IX(v) = (o~ + T2/2 + 6In v ) v and o-(v) = y v .

II. Hedging long-term options The absence of secondary markets for long-term foreign currency options prohibits the direct comparison of predicted versus traded option prices, but a test can be based on the accuracy of hedging strategies. For example, under the traditional model, foreign currency options can be replicated by a suitably chosen and constantly adjusted portfolio of domestic and foreign bonds. But if volatility is not constant, then this portfolio strategy will not replicate the option's payoffs. Our test of an option pricing model is based on the divergence between the paths of the predicted option price and the value of the replicating portfolio. This test has the advantage of not requiring data on actual traded option prices. To be specific, we consider a synthetic portfolio that balances a claim to the predicted option price with an offsetting position in the replicating portfolio. If the model is specified correctly, the synthetic portfolio will be selffinancing, that is, it will generate neither positive nor negative cash flows. But if the model is misspecified, the synthetic portfolio will generate random cash flows. The size of the cash flows generated by the synthetic portfolio over the life of the option provides a useful metric with which to evaluate the option pricing model. Indeed, the size of these cash flows may be of independent interest, even when data on traded option prices are available. To fix ideas, it is useful to consider in detail the behavior of the constant volatility model in the presence of misspecification. Suppose the exchange rate follows the process given in (1), but with volatility constant. The option price continues to satisfy the partial differential equation (3), but with all partials with respect to volatility suppressed. Using the boundary conditions appropriate for the type of option, the option price predicted by the constant volatility Journal of lnternational Money and Finance 1995 Volume 14 Number 3

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Pricing and hedging of long-term foreign currency options."A Melino and S M Turnbull

model can be expressed as (5)

C(t) - C(S(t); ~, T - t, EX, ro, rF),

where S(t) is the spot exchange rate, ~ is the level of constant volatility, T is the maturity date, E X is the exercise price, rD is the domestic risk free rate, and rF is the foreign risk free rate. Let B v and B D denote, respectively, the prices of foreign and domestic risk free bonds (to simplify notation, we will often normalize these prices to unity). We use C to distinguish the option price predicted by the constant volatility model from C, the true option price. If volatility is constant, the option can be replicated by a portfolio (or trading strategy) with time t value (6)

P(t) = D ( t ) S ( t ) B F ( t ) + Q(t)Bo(t),

where D denotes the number of foreign risk free bonds and Q is the number of domestic risk free bonds in the portfolio. In the replicating portfolio, the quantities of domestic and foreign risk free bonds are constantly adjusted to achieve (7)

D = Cs/BF,

<8>

O=(C-CsS)/B.

Let H denote the difference between the cumulative value of the trading strategy described by ( 6 ) - ( 8 ) and the option price predicted by (5). H ( t ) - fotD(r)d(Bv(r)S(r)) + foO(7) dBo(r) - (C(t) - t~(0)). Using It6's lemma and the above equations, gives (9)

dH=

[u2S 2 d t - (dS)2]Css/2.

Equation (9) measures the cash flow generated by a synthetic portfolio that balances a liability for the predicted option price with an offsetting position in the replicating portfolio. A test based on the lack of self-financing of this synthetic portfolio can be interpreted as an economically interesting way of describing the misspecification of (dS) 2. If the constant volatility model is true, then dH ~ O. In general, H(t) is a random quantity that measures the integral of the cash flows generated by the synthetic portfolio from the date of issue of the option to time t. E(H(T)) is the average payoff to holding the synthetic portfolio. It will reflect both pricing errors and market returns for bearing risk. We will say that the model is 'accurate' if E ( H Z ( T ) ) = 0; this requires the cumulative cash flows to be zero at maturity but allows for random cash flows over the life of the option. Other features of the path of H may also be of interest. We pursue this further in Section V. Discrete time counterparts to (9) are interesting for several reasons. First, although simulations can exploit (9), tests with real world data must limit their data requirements to discrete observations on the exchange rate process. Second, market participants cannot continuously rebalance the replicating portfolio. It is interesting to investigate the interaction of possible misspecifi378

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Pricing and hedging of long-term foreign currency options: A Melino and S M Turnbull

cation with discrete revision of the replicating portfolio. Finally, market participants often use the constant volatility model to price options but with estimates of volatility that are adjusted as frequently as the mix in the replicating portfolio. To what extent can such use of implicit volatilities temper the effects of misspecification? This is an important question and it is most easily addressed in discrete time. Suppose options are priced at discrete points in time according to (5). Define P(u, t) for u > t to be the value of the replicating portfolio at time u, given the portfolio was last revised at time t. The quantities of domestic and foreign bonds are chosen according to (7) and (8) at time t. In particular, for At>_0 (10)

P(t + At, t) = D ( t ) e x p ( r F A t ) S ( t + At) + Q(t)exp(r D At).

The cash flow generated by the synthetic portfolio with revision after At periods is (11)

AH(t)=[P(t+At,

t)-P(t,t)]-[C(t+At)-C(t)]

= P ( t + At, t) - C(t + At). As in the continuous time case, E ( H ( T ) ) measures the average total of the cash flows generated by the synthetic portfolio over the life of the option. If E [ H ( T ) 2] = O, then positive and negative cash flows always exactly offset each other over the life of the option. Because the replicating portfolio is not rebalanced continuously, the cash flows in (11) may be non-zero even if the constant volatility model is true. An argument due to Leland (1985) shows that, in the absence of misspecification, the size of the cash flows in (11) can be made arbitrarily small by using a suitably chosen revision period. 4 Some evidence on the size of these cash flows for economically relevant revision periods is provided below. IlL Simulations The accuracy of the constant volatility option pricing model is evaluated using four simulations. In the first experiment, volatility is a constant and the model is correctly specified. 5 The cash flows generated by (11) are computed for a variety of revision periods. This experiment allows us to gauge the size of the random cash flows generated by the synthetic portfolio that are due only to the discrete revision of the replicating portfolio. 6 In the remaining three simulations, volatility is stochastic. The second and third simulations investigate the consequences of incorrectly using the constant volatility model with an implicit estimate of volatility to price the option. The second simulation examines the consequences of this misspecification on the cash flows in (11) (delta hedging). Many market participants use the constant volatility model with implicit volatilities, but include an additional traded asset in the replicating portfolio as an ad hoc hedge against changes in volatility (delta plus vega hedging). Our third simulation examines the efficacy of this hedging strategy. We find that using the traditional model with implicit volatilities leads to major errors in Journal of International Money and Finance 1995 Volume 14 Number 3

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Pricing and hedging of long-term foreign currency options: A Melino and S M Tumbull

estimating the derivative of the option price with respect to volatility (the vega value). These errors, in turn, lead to inaccurate option prices and poor hedging performance. In the last simulation, we use the true stochastic volatility model to calculate the correct vega value and show that discrete revision alone does not diminish the efficacy of the delta plus vega hedging strategy. 111.4 C o n s t a n t volatility

In this simulation, exchange rates were generated by equation (1) with v constant. Melino and Turnbull (1991) estimate this model by maximum likelihood, and we use their estimate of volatility. For simplicity, the coefficients a and b are set to zero. We fixed the values for the domestic and foreign interest rates at their average values over the same period used in the estimation. For each simulated path of exchange rates, we priced European call options with maturities of 90 days, 1, 2.5, and 5 years. The replicating portfolio was revised at fixed intervals and the total hedging error ( E A H ) over the life of the option recorded. This was repeated over 1000 different sample paths. The mean square error of the total hedging error over the 1000 sample paths are reported in Table 1. The initial values of the options are also shown. 7 Three observations arise from Table 1. First, for all maturities, the mean total hedging error is small compared to the initial option value. Second, as the length of the revision period increases, the m e a n square error also increases. Third, the m e a n total hedging error fluctuates in sign. These results show that with constant volatility it is possible to replicate call options even with discrete trading, and the accuracy increases as the revision period decreases. TABLE1. Delta hedging--mean total hedging error: constant volatility. Revision period (days) 1 2 7 14 Option value

Maturity 90 days

I year

2.5 years

5 years

- 0.0041 (0.0046) 0.0003 (0.0072) - 0.0020 (0.0250) 0.0040 (0.0463) 0.55

0.0005 (0.0036) - 0.0016 (0.0075) 0.0051 (0.0246) - 0.0051 (0.0464) 0.85

- 0.0026 (0.0035) - 0.0071 (0.0069) - 0.0103 (0.0249) - 0.0126 (0.0440) 0.92

0.0009 (0.0030) - 0.0020 (0.0057) - 0.0059 (0.0192) - 0.0143 (0.0411) 0.77

Figures in parentheses are mean square errors. Initial exchangerate 82 U.S. ¢/C$ Exercise price 82 U.S. ¢/C$ Domestic interest rate 8.19percent Foreign interest rate 9.54 percent Standard deviation 0.0022 per day Number of simulations 1000 380

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III.B Stochastic

volatility

For the remaining three simulations, we used a discrete-time approximation to equations (1) and (4). The simulated exchange rates were generated according to St -

St - 1 =

( a + b S t _ 1) + vt - 1St - i et,

and v , - v , _ 1 = v t _ i ( a + y 2 / 2 + 6In v , _ l ) + y v t _ l F l t , where a, b, u, y and 6 are constants; and {(g,, ~,)} is a sequence of independent bivariate standard normal random variables, with correlation coefficient p. This model was estimated by Melino and Turnbull (1990). We use their estimated values for the parameters in the simulation. Again, for simplicity, the drift parameters a and b are set equal to zero. We used the same values for the domestic and foreign interest rates as those used for Table 1, and considered the same four European options. In practice, options are priced using implicit volatilities. These volatilities are usually calculated using at-the-money traded options, which typically have maturities of the order of two to three months. These 'short-term volatilities' are then used to price long-term options using an option model that assumes volatility is constant. In the next two simulations, we investigate the size of the pricing and hedging errors that arise from this approach when volatility is actually stochastic. The final simulation simply confirms that discrete revision alone does not lead to large hedging errors, if we use the true option pricing model. Before reporting the simulation results, it is useful to look in detail at a specific example that illustrates the consequences of misspecification. First, the stochastic volatility model is used to price an at-the-money 60-day European call option. From this price, an implicit volatility is computed using the constant volatility option model. Then this implicit volatility is used in turn to price a series of European options with maturities ranging from 90 days to 5 years. For comparison, we price the same series of options using the stochastic volatility model. Table 2 reports the results. For maturities of 90 days and 1 year there are no differences in option prices or the delta values. Using implicit volatility leads to the same prices for these options as the true model. For options with maturities of 2.5 years and 5 years, the constant volatility model over-estimates the option prices. For options with maturity of 2.5 years, the pricing error varies between 4 to 6 percent and for 5 year maturity options the pricing error varies between 12 and 15 percent. This finding is of some importance. Often, a 'short-term' implicit volatility is used by practitioners to price the option and then a premium is added to this price to compensate for the risk of not knowing the true volatility. For the example considered in Table 2, the constant volatility model already overprices the option. For all maturities there are large differences in vega, which is the derivative of the option price with respect to volatility. The differences are to be expected, as the interpretation of the vega value is different for the two models. For the constant volatility model, the variance of the terminal exJournal of International Money and Finance 1995 Volume 14 Number 3

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Pricing and hedging of long-term foreign currency options: A Melino and S M Turnbull

TABLE 2. Comparison of constant volatility and stochastic volatility option prices, delta and vega values European call option. Exercise Price

Maturity 60 days CV

SV

90 days CV

SV

1 year CV

2.5 years SV

CV

SV

5 years CV

SV

80

1.88 0.88 112.77

1.89 1.87 1.88 1.83 0.87 0.82 0.81 0.56 17.64 186.50 18.85 545.82

81

1.08 0.71 211.67

1.10 1.13 0.71 0.65 31.67 277.08

1.15 0.65 27.66

1.33 1.32 1.30 1.24 1.05 0.92 0.46 0.45 0.33 0.30 0.21 0.17 570.31 13.62 765.55 6 . 9 1 799.79 3.29

82

0.51 0.45 248.29

0.51 0.46 38.16

0.60 0.45 30.62

0.92 0.36 551.99

0.93 1.02 0.97 0.36 0.27 0.25 13.28 724.36 6.58

0.87 0.18 751.06

0.76 0.15 3.10

83

0.18 0.18 0.22 0.22 185.12 27.69

0.26 0.27 0.25 0.25 241.77 24.39

0.62 0.27 496.26

0.63 0.78 0.75 0.27 0.22 0.21 12.01 665.79 6.08

0.71 0.15 695.44

0.63 0.13 2.89

0.58 0.13 635.27

0.52 0.11 2.65

84

0.05 0.07 89.17

0.59 0.44 300.99

1.81 1.64 1.55 1.26 1.10 0.54 0.38 0.35 0.24 0.20 13.02 785.08 7.06 839.33 3.43

0.05 0.10 0.10 0.40 0.40 0.59 0.07 0.11 0.11 0.19 0.20 0.18 12.73 145.18 14.31 415.53 10.15 595.07

0.57 0.17 5.47

In each cell, the first figure is the option price, the second figure is delta, and the third figure is vega. CV Constant Volatility SV Stochastic Volatility Spot rate U.S. 82¢/C$ Standard deviation of log spot rate 0.0022000per day Implied standard deviation 0.0023367 per day Price of risk - 0.15 Domestic risk free rate of interest 8.19 percent Foreign risk free rate of interest 9.54 percent 8C Delta 0-S 0C Vega 8~b-

change rate distribution is tr 2 T, w h e r e tr is the volatility, and T is the maturity of the option. T h e effect of a change in the spot volatility on the variance of the terminal distribution increases with the maturity of the option. So it is not surprising to observe in T a b l e 2 that the constant volatility m o d e l yields a vega that increases with maturity. F o r the stochastic volatility model, the variance of the terminal distribution only partially d e p e n d s u p o n the initial spot volatility. At o u r p a r a m e t e r values, volatility is m e a n reverting, so this d e p e n d e n c e is a decreasing function of maturity. H e n c e it is r e a s o n a b l e to except the v e g a value to be a decreasing function of maturity (there is o n e exception).

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Delta hedging. In Table 3, we examine the ability to hedge the traditional model with a short-term implicit volatility. Each 'day' in the simulation, the true model is used to price an at-the-money, 60-day European call option. From this option price, an implicit volatility is calculated using the constant volatility model, each time the portfolio is revised. This model and the implicit volatility are used to form the replicating portfolio for four European options with maturity ranging from 90 days to 5 years. Three observations can be made. First, the mean total hedging error is consistently positive. Positive errors reflect the return for holding a risky portfolio and are no reason to be complacent. To use a more familiar example, the writer of a call on an equity can always invest the entire proceeds in the stock. On average, this will pay a positive return, but this would not be considered an effective hedging strategy! Second, the size of the mean total hedging error and its mean square error increases with the maturity of the option. Third, the mean square error increases with the replicating period. The results for 90 day and 1 year options provide an important constrast to the results in Table 2, which show that the pricing error using implicit volatility is small. Table 3 shows that, while the pricing error may be small, the constant volatility model with implicit volatility leads to a poor hedging strategy when volatility is stochastic.

Delta and vega hedging. To form a replicating portfolio when volatility is stochastic requires an extra asset, as there is an additional source of risk. TABLE 3. Delta hedging implicit volatility--Mean total hedging error.

Revision period (days) 1 2 7 14

Option value

Maturity 90 days

I year

2.5 years

5 years

0.1334 (0.0368) 0.1338 (0.0409) 0.1330 (0.0589) 0.1267 (0.0774) 0.60

0.2533 (0.0932) 0.2546 (0.0997) 0.2522 (0.1188) 0.2464 (0.1502) 0.94

0.3736 (0.1851) 0.3723 (0.1884) 0.3797 (0.2102) 0.3730 (0.2245) 1.03

0.4404 (0.2561) 0.4423 (0.2599) 0.4462 (0.2752) 0.4421 (0.2799) 0.88

Figures in parentheses are mean square errors. Parameter values Initial exchange rate 82 U.S. ¢/C$ a = 0.0 Exercise price 82 U.S. ¢/C$ b = 0.0 Domestic interest rate 8.19 percent c~= -0.745 Foreign interest rate 9.54 percent 3, = 0.192 Initial standard deviation 0.0022 per day 6 = -0.116 Number of simulations 1000 p = -0.181

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Pricing and hedging of long-term foreign currency options: A Melino and S M Turnbull

This simulation looks at using the constant volatility model with implicit volatilities but with an additional asset in the replicating portfolio as an ad hoc hedge against changes in volatility (delta plus vega hedging). Each 'day' in the simulation, the true model is used to price an at-the-money, 60-day European option. From this option price, an implicit volatility is calculated each time the replicating portfolio is revised. Once again, we consider the problem of pricing and hedging four European options with maturity ranging from 90 days to 5 years. For each option, the replicating portfolio is constructed from domestic bonds, foreign bonds, and the same 60-day option used to estimate the implicit volatility. The details are given in the Appendix. From Table 2, we see that the traditional model leads to errors in vega that are large and increase with the maturity of the option. Before reporting the simulation results, it is useful to consider some examples and explore the consequences of these errors for the replicating portfolio. Table 4 contains the calculations for two examples. In Part A, the objective is to construct a portfolio to replicate the instantaneous return on a 90-day European at-the-money call option. Option prices, delta and vega values are taken from Table 2. The replicating portfolio is composed of three assets: the domestic riskless asset, the foreign riskless asset and a traded option. The investment in the traded option is determined by the vega for the 90-day option divided by the vega TABLE 4. Construction of replicating portfolio. Part A 90 day

N u m b e r of traded options Investment in foreign currency Investment in domestic riskless asset

Implicit volatility

Stochastic volatility

300.99/248.29 = 1.21

30.62/38.16 = 0.80

- 0.11

0.08

8.62

- 6.00

Part B 5 year

N u m b e r of traded options Investment in foreign currency Investment in domestic riskless asset

384

Implicit volatility

Stochastic volatility

751.06/248.29 = 3.02

3.10/38.16 = 0.08

- 1.18

0.11

96.19

- 8.52

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Pricing and hedging of long-term foreign currency options: A Melino and S M Turnbull

for the 60-day traded option. Using implicit volatilities the investment is to buy 1.21 traded options. However, the theoretically correct investment is to buy 0.80 traded options. The error in the vega value affects the investment in the two other assets in the replicating portfolio. Using implicit volatility, the investment in the foreign riskless asset is to borrow 0.11 units of the foreign asset, and the investment in the domestic riskless asset is to buy 8.62 dollars. When the stochastic volatility model is used, the investment in the foreign riskless asset is to lend 0.08 units and the investment in the domestic riskless asset is to borrow 6.00 dollars. In part B, the objective is to construct a portfolio to replicate the instantaneous return on a 5 year European at-the-money call option. Using implicit volatility, the replicating portfolio is constructed by investing in 3.02 traded options, borrowing 1.18 units of the foreign riskless asset, and lending 96.19 dollars in the domestic riskless asset. Using the stochastic volatility model, the replicating portfolio is constructed by investing in 0.08 traded options, lending 0.11 units of the foreign riskless asset, and for the domestic riskless asset borrowing 8.52 dollars. These two examples show that the replicating portfolio constructed from a constant volatility option model with implicit volatility can look very different from that of the true model. Table 5 shows that these differences lead to large pricing and hedging errors. In comparing Tables 3 and 5, three observations are important. First, for 90-day options the addition of the extra asset into the replicating portfolio produces a major improvement. Both the absolute size of the mean total hedging error and the mean TABLE 5. D e l t a a n d v e g a h e d g i n g - - M e a n

Revision period (days) 1 2 7 14 Option value

t o t a l h e d g i n g error: implicit volatility.

Maturity 90 d a y s

1 year

2.5 y e a r s

5 years

0.0328 (0.0066) 0.0309 (0.0084) 0.0213 (0.0154) 0.0168 (0.0174) 0.60

- 0.4580 (0.2755) - 0.4660 (0.3003) - 0.4862 (0.3981) - 0.4989 (0.5252) 0.94

- 2.1019 (5.0759) - 2.1047 (5.1975) - 2.1231 (5.8173) - 2.1273 (6.4583) 1.02

- 5.3031 (31.5289) - 5.3137 (31.9713) - 5.3570 (34.1965) - 5.3599 (36.0200) 0.88

Figures in parentheses are m e a n square errors. Parameter values Initial exchange rate 82 U.S. ¢ / C $ a = 0.0 Exercise price 82 U.S. ¢ / C $ b = 0.0 Domestic interest rate 8.19 percent a = -0.745 Foreign interest rate 9.54 percent ~/= 0.192 Initial standard deviation 0.0022 per day 6 = -0.116 N u m b e r of simulations 1000 p = - 0.181

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square error are smaller. But the reverse is true for 1, 2.5 and 5 year options. For these options the mean total hedging errors are negative, so the replicating portfolio under-estimates the value of the option. Second the absolute values of the mean total hedging error and the mean square error increase in size as maturity increases. For options of maturity 2.5 and 5 years, the absolute values of the mean total hedging errors are very large in comparison to the initial option values. Third, the introduction of an additional asset in the replicating portfolio causes a dramatic deterioration in the ability to replicate long-term options. The reason for this deterioration is found in Table 4. The error in calculating the vega value leads to over investment in the traded option and the degree of over investment increases with maturity. It also implies borrowing in the foreign currency instead of lending. We do not wish to claim that practitioners actually follow this hedging strategy (at least, not any that are still in business!). A better approximation to actual practice would say that long-term options are priced off a 'volatility curve'. Our simulations pertain to the case where the volatility curve is assumed to move up and down by the same amount at all maturities. Any ad hoc procedure that has volatilities for long-term options respond less than one-for-one to movements in short-term volatilites would lead to better hedging performance. However, there is no end to such ad hoc strategies, and no obvious way to choose between them, especially since the pricing model treats volatility as constant. Our particular choice may exaggerate the costs of misspecification, but it is simple and illustrates our general point that such misspecification often seems more important for hedging than for pricing errors.

Delta and vega hedging with the stochastic volatility model. The simulation in Table 6 repeats the experiment in Table 4, but using the true model to calculate the portfolio weights. This has two purposes. It shows that the large errors in Tables 2 and 4 are not due principally to the use of discrete revisions. With the correct portfolio weights, the cash flows generated by discretely revising the replicating portfolio are very small. In addition, the results in Table 6 form a useful benchmark against which to compare the performance of the stochastic volatility model with actual exchange rate data. IV. Results using exchange rate data Three of the experiments conducted in the last section were repeated using real exchange rate data. The first experiment examines the use of delta hedging. The second experiment looks at the use of delta plus vega hedging. In both cases the constant volatility model is used with implicit volatilities. The last experiment re-examines the use of delta plus vega hedging using the stochastic volatility model. The results were very similar to those obtained from our simulations, so only the results of the last experiment will be discussed in any detail. 386

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Pricing and hedging of long-term foreign currency options: A Melino and S M Turnbull TABLE 6. Delta and vega hedging stochastic v o l a t i l i t y - - M e a n total hedging error. Revision period (days) 1 2 7 14

Maturity 90 days

1 year

2.5 years

5 years

0.0212 (0.0025) 0.0196 (0.0032) 0.0137 (0.0062) 0.0128 (0.0067)

0.0299 (0.0034) 0.0299 (0.0039) 0.0301 (0.0056) 0.0252 (0.0184)

0.0464 (0.0062) 0.0463 (0.0077) 0.0513 (0.0089) 0.0522 (0.0111)

0.0537 (0.0087) 0.0544 (0.0086) 0.0620 (0.0093) 0.0634 (0.0102)

Figures in parentheses are mean square errors. Parameter values Initial exchange rate 82 U.S. ¢/C$ a = 0.0 Exercise price 82 U.S. ¢/C$ b = 0.0 Domestic interest rate 8.19 percent a = -0.745 Foreign interest rate 9.54 percent y = 0.192 Initial standard deviation 0.0022 per day 15= -0.116 Number of simulations 1000 p = -0.181

Our empirical work requires the following data: (i) spot exchange rates for Canada; (ii) spot volatilities; (iii) implicit volatilities, (iv) interest rates for Canada and the USA. We use daily exchange rate data for the period January 2, 1975 to December 10, 1986. The data are the mid-day interbank market (Canada) average of bid and ask rates quoted by dealers surveyed by the Bank of Canada. To get the daily estimates of volatility, which are required for the stochastic volatility option model, we used the Kalman Filter described in Melino and Turnbull (1990). The traditional model requires data on implicit volatility. But, foreign currency options have only been traded on organized exchanges since the end of 1983. This is too short a sample period for our purposes. To get around this problem, we use the stochastic volatility model to price an at-the-money, 60-day European option, and then use this price to calculate an implicit volatility. Over the 1977-85 period there were substantial changes in interest rates. As a first approximation, we ignore these changes and assume that we can purchase bonds that pay the same constant rates as those employed in the simulations. Thus the design methodology is similar to that employed in Section III. Starting January 5, 1977, a portfolio was formed to replicate an option of maturity T, where T was set equal to 90 days, 1, 2.5, and 5 years respectively. Different revision periods are used, and at the end of each revision period, A H, is calculated. When the option matures, the value of the total hedging error, EAH~, is recorded. A new portfolio is formed every 28 days and the whole process repeated. For options of maturity 90 days the last portfolio is Journal of International Money and Finance 1995 Volume 14 Number 3

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Pricing and hedging of long-term foreign currency options: A Melino and S M Turnbull

formed in June 1986, so that a total of 124 portfolios were formed. For options of maturity one year, the last portfolio is formed in September 1985, giving a total of 115 portfolios; for options of maturity 2.5 years, the last portfolio is formed in May, 1984, giving a total of 96 portfolios; for options of maturity 5 years, the last date is September, 1981, giving a total of 62. For options of a given maturity, after all the portfolios have been formed and the total hedging error over the life of each option determined, the mean and mean square error of the total hedging error are calculated. Thus for options of maturity one year, 115 observations were used in calculating these statistics. The use of over-lapping results implies that all empirical results must be treated with caution. Table 7 shows what happens when the stochastic volatility model is used to construct the portfolio weights. The most striking aspect of these results is the small size of both the mean total hedging error and the mean square error. Compared to the simulation results of Section III, there are several differences. First, in Table 6 the mean total hedging errors are small but all positive. This is not so in Table 7; options with maturity of greater than 90 days have negative mean total hedging errors. Also, the mean square errors are uniformly lower with the real data than with the simulation data.

V. Implications and limitations The pricing model developed in Section II ignores several features of practical importance. Financial institutions that offer long-term options to clients typically have three components in the price that they charge: (a) the initial value of the replicating portfolio - - often based on a model, such as that in Section II, that neglects transactions costs, (b) a fee for providing the brokerage a n d / o r trading services that are required to implement the hedging strategy, TABLE 7. Real data: delta and vega hedging--Mean total hedging errors: stochastic volatility. Revision period 1 2 7 14 Mean value of option

90 days Call Put 0.0150 (0.0020) 0.0157 (0.0025) 0.0065 (0.0025) - 0.0079 (0.0062) 0.58

1 year Call

Put

2.5 years Call Put

5 years Call Put

0.0130 -0.0173 -0.0188 -0.0316 -0.0326 -0.0247 -0.0253 (0.0020) (0.0025) (0.0026) (0.0035) (0.0036) (0.0027) (0.0027) 0.0133 -0.0187 -0.0195 -0.0310 -0.0317 -0.0224 -0.0230 (0.0024) (0.0025) (0.0025) (0.0034) (0.0035) (0.0026) (0.0026) 0.0067 - 0.0286 - 0.0279 - 0.0402 - 0.0396 - 0.0345 - 0.0335 (0.0025) (0.0038) (0.0038) (0.0038) (0.0038) (0.0032) (0.0031) - 0.0080 - 0.0538 - 0.0538 - 0.0489 - 0.0484 - 0.0412 - 0.0402 (0.0062) (0.0106) (0.0106) (0.0044) (0.0044) (0.0037) (0.0037) 0.81

0.91

1.86

1.02

3.15

0.90

4.44

Figures in parentheses are mean square errors. 388

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Pn'cing and hedging of long-term foreign currency options: A Melino and S M Tumbull

and (c) a fee to compensate for uncertainty arising from using a false model. Our discussion so far has concentrated entirely on (a). The impact of transactions costs on option pricing and hedging is an area of active research, and is discussed by Leland (1985), and Boyle and Vorst (1992), among others. Very little research has been devoted to the thorny question of how to price the uncertainty from not knowing the true model. Due to both ignorance and choice, option pricing models are inevitably false, so that the 'hedged' portfolios to which they lead still must be viewed as claims to a path of random cash flows. In practice, an important part of a long-term foreign currency option's price is compensation for bearing this uncertainty. Institutions deal with model uncertainty in fairly informal ways, as they must be given the paucity of theoretical guidelines. One approach is to set up a reserve fund. The reserve fund is used to provide any infusions of cash required to maintain the replicating portfolio over the life of the option. In principle, the size of the reserve fund is chosen so that there will be only a small chance that it will be exhausted (and additional funds committed). It is not clear how this 'change' should be calculated, but one approach is to use past data. Using an estimate of the reserve fund to measure the costs of model uncertainty has a number of shortcomings. In particular, it ignores the fundamental rule that only systematic risk should be priced. We know that there are volatile random cash flows that should have zero market price. On the other hand, because of the imperfect link between risk and variance, reserve fund estimates might under-estimate the cost of excessive exposure to risk. Nonetheless, standard asset pricing theory does not provide much guidance for decision-making with model uncertainty and both practitioners and regulators need some way to deal with this problem, s We take the random cash flows associated with a hedging strategy, and use their empirical distributions to estimate the size of the required reserve fund. Let M v denote the maximum value of - H ( t ) for t < T. Since H(0) = 0, we have M v > O. One way of making the reserve fund operational is to choose the value m~ such that P ( M r < m s ) = a . In this notation, m,, is the value of a reserve fund that covers the additional cash requirements of the hedge portfolio with probability 0.9. Table 8 reports the values of m s estimated from our TABLE 8. E s t i m a t e d size of the reserve fund. 90 day call option 1-day revision

Table Table Table Table Table

1 3 5 6 7

5-year call option 1-day revision

a = 0.5

tr = 0.9

ct = 0.99

a = 0.5

a = 0.9

a = 0.99

0.05 0,04 0.04 0.02 0.02

0.12 0.13 0.10 0.06 0.06

0.18 0.24 0.16 0.10 0.10

0.04 0.05 5.61 0.03 0.05

0.09 0.16 7.90 0.09 0.10

0.14 0.31 9.79 0.15 0.14

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previous simulations, for various choices of a. In many cases the value of m~ is large compared to the predicted model price. Comparing to Table 2, we see that the pricing errors that arise from using the wrong model are small relative to the cost of the hedging errors. One obvious source of misspecification in our work deserves specific mention. All the option pricing models used in this paper have assumed that interest rates--domestic and foreign--are constant over the life of the option. Melino and Turnbull (1990) present some evidence that this is not a serious problem for traded options. This is confirmed by Amin and Bodhurtha (1991), who also present evidence that long-term option prices can be significantly affected by stochastic interest rates. One approach that takes into account the effect of stochastic interest rates is to use the hedging strategies based on the constant term structure assumption, but then measure the cash flows to the hedged portfolio using market prices for domestic and foreign bonds. In some preliminary work, we found that this led to much larger random cash flows and therefore larger reserve funds than those reported in Table 8. A second approach is to extend the option pricing model to include stochastic interest rates, as in Amin and Jarrow (1991), and follow the hedging strategy associated with this extended model. While this can be done using Monte Carlo techn i q u e s - s e e Rubinstein (1991)--it is computationally expensive. Some compromise between descriptive ability and mathematical complexity is required, and we hope that our results based on constant interest rates provide a useful first approximation.

VI. Conclusions and discussion This study has examined the effects that stochastic volatility has upon the pricing and hedging of long-term options. Two conclusions can be drawn. First, a constant volatility option pricing model with a 'short-term implicit volatility' does a poor job in pricing long-term options. In our experiments, the constant volatility model over estimated option prices. Second, the traditional model leads to bad hedging strategies, even in those cases where the prices are approximately correct. The recognition that volatility is stochastic requires the addition of one more asset to the replicating portfolio. We find that the addition of such an asset improves the ability of the traditional model to replicate short-term options. However, the reverse is true for long-term options. The constant volatility option model over-estimates vega by an amount that increases with maturity. When volatility is mean reverting, the effects of the initial spot volatility on the terminal probability distribution decrease as the maturity of the option increases. The constant volatility model fails to take this into account and so pricing and hedging errors result. These errors increase in size as the option's maturity increases.

Appendix Assume the stochastic process for the exchange rate, S, is d S = i z s ( S ) dt + vS d Z s ,

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Pricing and hedging of long-term foreign currency options: A Melino and S M Turnbull

and for the volatility dv = i~v(v) dt + ~r(v) dZv,

where Z s and Z v are standard Wiener processes whose increments have instantaneous correlation p; tzs(S) is the drift for the exchange rate process; ~o(v) is the instantaneous mean and 0-2(v) is the instantaneous variance of the volatility process. We require that zero is an absorbing barrier for S. The objective is to construct a portfolio to replicate an option denoted by C. This can be done by purchasing D units of the risk free foreign bond, Q units of the risk free domestic bond, and some other traded options on the same underlying exchange rate. The value of the total portfolio is given by V =- C ( S , v, T - t) - [DS + Q + n c T ( s , V, T~ - t)],

where C T is a traded option that matures at time Ta; exercise price is EXa; and n is the number of traded options purchased. The initial value of the total portfolio is zero. Differentiation yields oC oC d V = L[ C ] dt + - ~ v S d Z s + -ff-ff a dZ~ i

- I D[ tz(S) + Sr F ] dt + DvS d Z s + Qr dt

+n L[C r] dt + ~ v a d Z

s + - g 7 0 - dZ°

'

where O 3 O L-~ - ~ + l Z s - ' ~ + l z v ~ ' ~ +

o2S 2 32 O2 0-2 32 2 ~0S2 +uSP0-o--S~+--f-Ov--"£"

In order to eliminate the risk arising from d Z s and dZo, pick D and n such that 3C 3C T 0-S- = D + n c~S ' o~C ~C r - - ~ - n - -

Ov

3V '

and to eliminate arbitrage profits, pick Q so that C = D S + Q + n C y.

Notes 1. See McFarland et al. (1982), Hsieh (1988), Boothe and Glassman (1987) and Tucker and Scott (1987). 2. The effects of stochastic volatility upon stock and index options are examined by Hull and White (1987a), Johnson and Shanno (1987), Scott (1987) and Wiggins (1987). 3. To assume a reflecting barrier would imply the existence of arbitrage opportunities, if it is possible to reach the barrier. We thank John Powell for this insight. 4. Leland's (1985) analysis can be extended to cover the class of processes defined by (1). 5. Grabbe (1983) describes the traditional option model. 6. For an alternative approach to revising the portfolio, see Asay and Edelsburg (1986). They suggest revising the portfolio when the difference in the value of the underlying asset and portfolio reaches a specified size. 7. The option values reported at the bottom of Table 1 use the data values listed as inputs. Thus a 90-day, at-the-money European call option is priced at 0.5527¢. Note that the option's value does not always increase as the maturity increases. For common stocks, in Journal of lnternational Money and Finance 1995 Volume 14 Number 3

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Pn'cingand hedgingof long-termforeign currency options:A Melino and S M Turnbull the absence of dividends, a call option's value is an increasing function of maturity. This is not necessarily true for foreign currency options. 8. Group of Thirty (1993) discuss the need to measure the size of reverse funds required to compensate for exposure to various risks not included in the usual academic discussion, such as risk arising from model uncertainty. References

AMIN, KAUSHIK I. AND JAMES N. BODURTHA, 'On valuing international money market contingent claims: stochastic interest rates and the American exercise feature,' University of Michigan Working Paper, 1991. AMIN, KAUSHIK I. AND ROBERT A. JARROW, 'Pricing foreign currency options under stochastic interest rates,' Journal of International Money and Finance, September 1991, 10: 310-329. ASAY, MICHAELAND CHARLESEDELSBERG,'Can a dynamic strategy replicate the returns of an option?' Journal of Futures Markets, Spring 1986, 6: 63-70. BODHURTHA,JAMESN. AND GEORGESR. COURTADON,'Tests of the American option pricing model on the foreign currency-options market,' Journal of Financial and Quantitative Analysis, June 1987, 22: 153-168. BOOTHE, PAUL AND DEBRA GLASSMAN, 'The statistical distribution of exchange rates,' Journal of International Economics, May 1987, 22: 297-319. BOYLE, PHELIM P. AND TON VORST, 'Option replication in discrete time with transaction costs,' Journal of Finance, March 1992, 47: 271-293. COOPER, IAN, COSTAS KAPLANIS,ANTHONYNEWBERGER AND STEPHEN SCHAEFER, 'Option hedging,' London Business School Working Paper, June 1986. COURTADON, GEORGES R., 'The pricing of options on default free bonds,' Journal of Financial and Quantitative Analysis, March 1982, 17: 75-100. Cox, JOHN C., JOHATHANE. INGERSOLLAND STEPHENA. ROSS, 'An intertemporal general equilibrium model of asset prices,' Econometrica, March 1985, 53: 363-384. GRABBE, J. ORL1N, 'The pricing of call and put options on foreign exchange,' Journal of International Money and Finance, December 1983, 2: 239-254. GROUP OF THIRTY, Derivatives: Practices and Principles, Report prepared by the Global Derivatives Study Group, Washington DC, July 1993. HANSEN, EARS PETER, 'Large sample properties of generalized method of moments estimators,' Econometrica, July 1982, 50: 1029-1054. HSIEH, DAVID A., 'The statistical properties of daily exchange rates: 1974-1983,' Journal of International Economics, February 1988, 241: 129-145. HULL, JOHN AND ALAN WHITE, 'The pricing on assets with stochastic volatilities,' Journal of Finance, June 1987, 42:281-300 (1987a). HULL, JOHN AND ALAN WHITE, 'Hedging the risks from writing foreign currency options,' Journal of International Money and Finance, June 1987, 6:131-152 (1987b). JOHNSON,HERB AND DAVIDSHANNO,'Option pricing when the variance is changing,' Journal of Financial and Quantitative Analysis, June 1987, 22: 143-151. LELAND, HAYNE E., 'Option pricing and replicating with transactions costs,' Journal of Finance, December 1985, 40: 1283-1301. MCFARLAND, JAMES W., R. RICHARDSONPETTIT AND SAM K. SUNG, 'The distribution of foreign exchange price changes: trading day effects and risk measurement,' Journal of Finance, June 1982, 37: 693-715. MELINO, ANGELO AND STUART M. TURNBULL, 'Pricing foreign currency options with stochastic volatility,' Journal of Econometrics, July/August 1990, 45: 239-265. 392

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MELINO, AYGELOAND S. M. TURNBULL,'The pricing of foreign currency options,' Canadian Journal of Economics, May 1991, 24: 251-281. PkESS, JAMES S., 'A compound events model for security prices,' Journal of Business, July 1968, 40: 317-335. RUBINSTEIN,REUVEN, Simulation and the Monte Carlo Method, New York: Wiley, 1991. SCOTF, LOUIS D., 'Option pricing when the variance changes randomly: theory, estimators, and application,' Journal of Financial and Quantitative Analysis, December 1987, 22: 419-438. SHASTRI, KULDEEP AND KISHORE TANDON, 'Valuation of foreign currency options: some empirical tests,' Journal of Financial and QuantitatioeAnalysis, June 1986, 21: 145-160. TUCKER, ALAN L. AND ELTON ScoTr, 'A study of diffusion processes for foreign exchange rates,' Journal of International Money and Finance, December 1987, 6: 465-478. WIGGINS,JAMESB., 'Option values under stochastic volatility: theory and empirical estimates,' Journal of Financial Economics, December 1987, 19: 351-372.

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