pricing currency options in the presence of time-varying volatility and non-normalities

J. of Multi. Fin. Manag. 16 (2006) 291–314

Pricing currency options in the presence oftime-varying volatility and non-normalities

G.C. Lim a,∗, G.M. Martin b, V.L. Martin a

a Department of Economics, University of Melbourne, 3010 Vic., Australiab Department of Econometrics and Business Statistics, Monash University, Australia

Received 25 August 2005; accepted 28 August 2005Available online 30 September 2005

Abstract

A new framework is developed for pricing currency options in the case where the distribution of exchangerate returns exhibits time-varying volatility and non-normalities. A forward-looking volatility structure isadopted whereby volatility is expressed as a function of currency returns over the life of the contract. Time tomaturity and moneyness effects in volatility are also modelled. An analytical solution for the option price isobtained up to a one-dimensional integral in the real plane, enabling option prices to be computed efficientlyand accurately. The proposed modelling framework is applied to European currency call options for the UKpound written on the US dollar, over the period October 1997–June 1998. The results show that pricinghigher order moments improves both within-sample fit and out-of-sample prediction of observed optionprices, as well as having important implications for constructing hedged portfolios and managing risk.© 2005 Elsevier B.V. All rights reserved.

JEL classification: C13; G13; F37

Keywords: Option pricing; Skewness and fat-tails; Time-varying volatility; Generalized Student t distribution; Semi-non-parametric distribution

1. Introduction

One of the main challenges in devising empirical models of currency options is to spec-ify models that are rich enough to capture the empirical characteristics commonly observed incurrency returns, whilst being computationally simple to evaluate. Well-established empirical

∗ Corresponding author. Fax: +61 3 8344 6899.E-mail address: [email protected] (G.C. Lim).

1042-444X/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.mulfin.2005.08.004

292 G.C. Lim et al. / J. of Multi. Fin. Manag. 16 (2006) 291–314

characteristics of currency returns distributions are excess kurtosis and skewness, as well as time-varying moments (e.g. de Vries, 1994 and Jondeau and Rockinger, 2003). None of these empiricalcharacteristics are priced in the Garman and Kohlhagen (1983) model of currency options, whichassumes normal currency returns and constant volatility. In particular, it is well known that theGarman and Kohlhagen (GK) model misprices options, as evidenced by the occurrence of impliedvolatility smiles (e.g. Melino and Turnbull, 1990 and Bollen and Rasiel, 2003). To correct formispriced currency options across the moneyness spectrum, more complex option price modelshave been developed, which take into account the observed features of currency returns. Somerecent examples are: Melino and Turnbull (1990) and Guo (1998), which allow for stochasticvolatility; Bates (1996), which includes both jump processes and stochastic volatility; Lim et al.(1998), which allows for a generalized autoregressive conditional heteroscedasticity (GARCH)structure for volatility and non-normal currency returns; and Bollen et al. (2000), which adoptsa Markovian switching framework. For a recent analysis of currency options that compares theempirical performance of some of the alternative time-varying moment models that have beenproposed, but with the assumption of conditional normality maintained, see Bollen and Rasiel(2003).

The approach adopted in this paper is to specify directly the risk-neutral probability distributionof the exchange rate at the time the option contract matures, with the parameters of the distributionrepresenting the risk-neutralised versions of their empirical analogous. To allow for time-varyingvolatility, a volatility structure based on the work of Rosenberg and Engle (1997) and Rosenberg(1998) is adopted. This structure captures forward-looking behaviour, with the volatility specifiedas a function of currency returns over the life of the contract. This is in contrast to the backward-looking stochastic volatility and GARCH models, whereby volatility is expressed as a function oflagged currency returns. The Rosenberg and Engle (RE) volatility specification is also extendedhere to allow for additional determinants that capture the time series patterns commonly observedin implied volatilities. The choice of the determinants is influenced by the work of Dumas et al.(1998), who allow for maturity and strike price effects, and Ait-Sahalia (2002), who considersmoneyness as a determinant of the volatility. See also Kim and Kim (2003) for additional examplesof determinants of volatility.

An important feature of the proposed modelling framework is that the currency option price isexpressed as a one-dimensional integral in the real plane. In the special case of normal currencyreturns and constant volatility, the integral is standard, being based on the lognormal distribution.For the more general case of time-varying volatility and non-normalities, the integral is still one-dimensional, but needs to be computed numerically on the real plane. This is in contrast to modelsthat rely on Monte Carlo methods to price options, which can be computationally demanding(for example, Hafner and Herwartz, 2001 and Lehar et al., 2002) and potentially less accurateunless a very large number of simulations are used. Even in the case of the stochastic volatilitymodel proposed by Heston (1993) and the GARCH time-varying volatility model proposed byHeston and Nandi (2000), the computational requirements are still more demanding than thepricing framework proposed here, as integration needs to take place over the complex plane.Moreover, these models still assume normal returns, and thus lack the flexibility of the modellingframework developed here. Bollen and Rasiel (2003) use a lattice approach to price currencyoptions for alternative time-varying moment specifications, whilst still assuming conditionallynormal currency returns.

In specifying the risk-neutral distribution two candidates are used to capture the non-normalitiesin the conditional distribution of currency returns: the generalized Student t distribution (GST here-after) of Lye and Martin (1993, 1994) and the semi-non-parametric distribution (SNP hereafter)

G.C. Lim et al. / J. of Multi. Fin. Manag. 16 (2006) 291–314 293

of Gallant and Tauchen (1989). The use of the GST distribution is motivated, in part, by the sim-ulation experiments in Lim et al. (1998), which demonstrate that this distribution has sufficientflexibility to price currency options. The SNP distribution has recently been adopted to modelthe dynamics of asset markets by Chernov et al. (2003) and represents an alternative to the GSTdistribution for capturing non-normalities in returns data. Both the GST and SNP distributionsare applied in the context of pricing equity options by Lim et al. (2005). As highlighted therein,the SNP option pricing model is related to the option pricing model proposed by Jarrow and Rudd(1982) and applied by Corrado and Su (1997) and Capelle-Blancard et al. (2001), amongst others.However, in contrast to the Jarrow and Rudd model, the SNP pricing method ensures non-negativeestimates of the risk-neutral probabilities and, hence, is a valid model for the pricing of currencyoptions; see Lim et al. (2005) for more on this point.1

The modelling framework is applied to the pricing of European currency call options for theUK pound written on the US dollar over the period October 1997–June 1998. A range of within-sample and out-of-sample experiments are performed, in order both to compare the differentdistributional specifications and to assess the gains to be had by generalizing beyond the GKmodel for currency options. The results show strong evidence of time-varying volatility over thesample period as well as significant non-normalities in the conditional distribution of currencyreturns, thereby validating the more general pricing approach. In addition, they demonstrate thatthe option price model in general performs better when the risk-neutralised distribution is specifiedas the GST distribution rather than as the SNP distribution. These implications are investigatedfurther in the context of devising dynamic hedging strategies to minimize the portfolio risk ofexposure to exchange rate movements.

The rest of the paper proceeds as follows. Section 2 presents the option pricing framework thatcombines time-varying volatility and a flexible risk-neutral probability distribution specificationto capture non-normalities in currency returns. A maximum likelihood procedure is proposed inSection 3 to estimate the parameters of the model. The empirical performance of the competingmodels is investigated in Section 4 using both within-sample and out-of-sample criteria. Theimplications of non-normalities and time-varying volatility for constructing hedged portfoliosand managing risk are also discussed. Conclusions are presented in Section 5.

2. The option pricing model

In this section a pricing model is developed, using a risk-neutral probability distribution thathas sufficient flexibility to capture the empirical characteristics of currency returns. A specialfeature of this model is that it nests a number of pricing models, including the GK option pricingmodel, which is equivalent to the Black and Scholes (1973) model for pricing European calloptions on equities paying a continuous dividend stream equal to the foreign interest rate.

The approach adopted in the paper is to specify the risk-neutral distribution directly, such thatall parameters specified and estimated are risk-neutralised quantities. Associated with these risk-neutralised parameters are the parameters of the underlying empirical distribution of currencyreturns, which, in principle, could be used to extract an estimate of the risk preferences, factoredinto the option prices. To this extent the approach is similar to the method adopted by Jarrow

1 Other methods that could be used to incorporate the empirical features of currency returns in option pricing, in additionto those already mentioned in the text, include binomial trees (Jackwerth and Rubinstein, 1996) and non-parametric kernelmethods (Ait-Sahalia and Lo, 1998 and Ghysels et al., 1998). These methods are not pursued in the present paper.


and Rudd (1982) and applied subsequently by Corrado and Su (1997) and Capelle-Blancard etal. (2001), with the risk-neutral distribution specified as a Gram-Charlier distribution; see alsoBollen and Rasiel (2003). Given that the focus of the present paper is on the pricing of options,only parameter estimates associated with the risk-neutral probability distribution are reported.

Let St be the spot exchange rate at time t of one unit of the foreign currency measured in thedomestic currency. Define rt and it as the respective domestic and foreign risk free annualizedinterest rates at time t for maturity at time T. Consider writing a European call option at time t onSt with strike price X, that matures at time T. The price of the currency option is

F (St) = Et[e−rtτt max(ST − X, 0|St)]

= e−rtτt

∫ ∞

X

(ST − X)g(ST |St)dST ,(1)

where Et denotes the expectation formed at time t with respect to the risk-neutral distribution forthe exchange rate at the time of maturity, g(ST|St), and τt = (T − t)/252 is the maturity of the optioncontract expressed as a proportion of a (trading) year. In this paper, the risk-neutral distributionfor the continuously compounded return over the life of the option is assumed to be representedby

ln

(ST

St

)=(

rt − it − σ2T |t2

)τt + σT |t

√τtzT , (2)

where zT is a zero mean, unit variance random variable that represents unanticipated movements inexchange rates and that is uncorrelated with rt − it − σ2

T |t/2, and σT|t is the conditional volatility,the specification of which is detailed in the following section. The specification of the distribu-tion of zT determines, in combination with the specification of σT|t, the form of the risk-neutraldistribution used to price the currency option in (1).

2.1. Conditional volatility

The two models most commonly adopted to capture time-variation in volatility when pricingcurrency options are the stochastic volatility model of Heston (1993), applied by Guo (1998)and Chernov and Ghysels (2000), amongst others, and the GARCH model of Engle and Mustafa(1992), applied by Lim et al. (1998), Hafner and Herwartz (2001) and Martin et al. (2005), amongstothers. In both classes of models, the volatility specification is backward-looking as it is laggedreturns that influence current volatility.

An alternative approach, which is adopted here, is to allow the volatility structure to be forward-looking, with the volatility specified to be a function of the return over the remaining life of thecontract. This framework was originally proposed by Rosenberg and Engle (1997) and appliedby Rosenberg (1998) and Lim et al. (2005).

The RE volatility specification is given by

ln σT |t = β0 + β1 ln

(ST

St

), (3)

which shows that the natural logarithm of volatility is a linear function of the currency return overthe life of the option contract, namely ln(ST/St). As the remaining time until the contract maturesdecreases, St approaches ST. On the day that the contract matures, t = T, the volatility reducesto σT|t = exp (β0). An important feature of the RE volatility model is that, in common with a


GARCH-type model but in contrast with a stochastic volatility model, there is no additionalrandom error term. The randomness in volatility arises from the randomness in the spot price atthe time of maturity, ST, with this future price being unknown at time t. This means that the optionmarket is complete, with there being, as a consequence, no need to resort to additional equilibriumarguments in order to price options, in contrast with the case for volatility specifications based onstochastic volatility.

The volatility specification adopted in the present paper extends the RE specification above toinclude functions of maturity (τt = (T − t)/252) and moneyness (X/St),

ln σT |t = β0 + β1 ln

(ST

St

)+ β2τt + β3τ

2t + β4 ln

(X

St

)+ β5

(ln

(X

St

))2

. (4)

The terms τt and τ2t in (4) allow for deterministic time-variation, whereby volatility is a function

of the time to maturity. This specification is motivated by the work on volatility functions by Dumaset al. (1998) and is adopted in the empirical analysis conducted below to help capture the observeddependence on maturity of implied volatility over the sample period investigated. The remainingterms, ln (X/St) and (ln (X/St))2, are included to allow for the effects of moneyness on volatility,as motivated by the work of Ait-Sahalia (2002).

2.2. The risk-neutral distribution

Given (2) and (4), the risk-neutral probability distribution at the time of maturity, g(ST|St), isgiven by

g(ST |St) = |J |p(zT ), (5)

where J is the Jacobian of the transformation from zT to ST, defined as

J = dzT

dST

=1 + β1σ

2T |tτt − β1

(ln(

STSt

)−(

rt − it − σ2T |t2

)τt

)ST σT |t

√τt

. (6)

As is clear from (6), the allowance for time-varying volatility in the option price formulationis not associated with any additional computational complexity. The specification in (4), beinga function of ST, enables a closed form solution for the Jacobian in (6) to be produced. Thiscontrasts, for example, with the stochastic volatility option pricing model of Heston (1993),in which a solution for the option price is produced only under the assumption of conditionalnormality. Moreover, the solution that is produced in the case of the Heston model is analyticalonly up to two one-dimensional integrals in the complex plane, a situation that contrasts with thesingle real integral whose evaluation is required in (1).2

The computational simplicity associated with the use of the time-varying volatility structure in(4) differs from the computational burden associated with the use of a GARCH-type specification.Although the GARCH option price of Heston and Nandi (2000) produces a closed form solution,up to a single complex integral, the augmentation of the GARCH model with a non-normalconditional distribution (e.g. Hafner and Herwartz, 2001, Lehar et al., 2002 and Martin et al.,

2 Estimation of Heston-type models using observed option prices also has the additional computational burden associatedwith the presence of the latent volatilities; see, for example, Guo (1998), Bates (2000) and Chernov and Ghysels (2000).


2005) entails the use of computationally intensive Monte Carlo simulation for the option priceevaluation.

2.3. Alternative distributional forms for zT

It is well known that the distribution of currency returns exhibits both fat-tails and skewness;see, for example, de Vries (1994) and more recently Jondeau and Rockinger (2003). This is stilltrue even when time-varying conditional moments are used (Bai et al., 2003). To capture theseempirical features, zT in (2) is assumed to have a non-normal distribution. Two types of non-normal distributions are considered: the GST distribution of Lye and Martin (1993, 1994), andthe SNP distribution of Gallant and Tauchen (1989). Whilst there are many other choices thatcould be contemplated for the risk-neutral probability distribution, these specifications both sharethe advantage of a convenient nesting of the lognormal distribution and, as a consequence, theGK currency option price model.3

2.3.1. Generalized Student tThe general form of the GST distribution is given by (Lye and Martin, 1993)

p(zT ) = k−11 σw exp

[θ1 tan−1

(µw + σwzT√

ν

)+ θ2 ln(ν + (µw + σwzT )2)

+4∑

j=1

θj+2(µw + σwzT )j

⎤⎦ , (7)

where µw and σw are chosen to ensure that zT is standardized to have zero mean and unit varianceand, k1 is the normalizing constant, defined by

k1 =∫

p(zT )dzT . (8)

The moments of the distribution in (7) exist as long as the parameter on the highest even-orderterm, namely θ6 in (7), is negative. The power term, (ν + (µw + σwzT )2)

θ2 , is a generalizationof the kernel of a Student t density and the parameters in that term, ν and θ2, along with theparameters θ4 and θ6, control the degree of kurtosis in the distribution. The odd moments of thedistribution, including functions of them such as skewness, are controlled by the parameters θ1,θ3 and θ5.

The price of the currency option as defined by (1)–(8) nests a number of special cases. Setting

β1 = β2 = β3 = β4 = β5 = 0, (9)

in (4) results in a constant volatility model with non-normal currency returns. Further imposingthe restrictions

θ4 = −0.5, and θj = 0, ∀j = 4, (10)

3 In an earlier version of the paper we also considered a lognormal mixture distribution for zT. This model did notimprove upon the performance of the SNP model in the empirical application, and hence has been eliminated from thecurrent version. The generalized Student t distribution investigated by Jondeau and Rockinger (2003) is another possiblecandidate for the distribution of zT.


in (7) yields the GK model, as p(zT) reduces to the standardized normal distribution and g(ST|St)in (5) becomes lognormal. In this special case of constant volatility and normal currency returns,the GK option price is

FGKt = Ste

−itτtΦ(d1) − Xe−rtτtΦ(d2), (11)

where

d1 =ln

(St

X

)+(

rt − it − σ2

2

)τt

σ√

τt

,

d2 =ln

(St

X

)+(

rt − it − σ2

2

)τt

σ√

τt

,

(12)

Φ(·) is the cumulative normal distribution function, and σ = exp(β0), with β0 as given in (4). Inthe empirical analysis conducted below, the form of the GST distribution specified for zT is

p(zT ) = kσw exp

[θ1 tan−1

(µw + σwzT√

ν

)−(

1 + ν

2

)ln(ν + (µw + σwzT )2)

− 0.5(µw + σwzT )2], (13)

where θ1 controls skewness and ν models the degree of kurtosis in the distribution. The specifi-cation of the negative sign on the fourth order polynomial term ensures that all moments of thedistribution exist, whilst the magnitude of 0.5 is chosen in order to ensure that p(zT) nests thenormal distribution.4

Using (13) in (5), which, in turn, is substituted into (1), gives the GST currency option price,

FGSTt = e−rtτt k1

∫ ∞

X

(ST − X)σw exp

[θ1 tan−1

(µw + σwzT√

ν

)

−(

1 + ν

2

)ln(ν + (µw + σwzT )2) − 0.5(µw + σwzT )2

]|J |dST , (14)

where J is defined by (6) and zT is defined by rearranging (2).

2.3.2. Semi-non-parametricThe form of the SNP distribution adopted here is based on augmenting a normal density

for the standardized variable zT in (2) by a polynomial that captures higher order moments.Jarrow and Rudd (1982) were the first to adopt this approach, which has more recently beenimplemented by Corrado and Su (1997) and Capelle-Blancard et al. (2001). As the adopteddistribution is based on a Taylor series expansion around the GK option price distribution, thatis, the lognormal distribution, Jarrow and Rudd interpret this distribution as a local risk-neutraldistribution.

4 Strictly speaking the normal distribution is nested by (13) only when θ1 = 0 and the coefficient on the logarithmic termis set to zero also. This generalization of the Student t distribution circumvents the problems identified by Duan (1999) inmodelling nonnormalities in option prices using the Student t distribution.


A problem that arises in using the Jarrow and Rudd (1982) specification is that the generatedprobabilities are not constrained to be non-negative (Lim et al., 2005). To impose non-negativityon the underlying risk-neutral probability distribution, an alternative specification is used here,based on the SNP density of Gallant and Tauchen (1989). The form of the standardized returnsdistribution is

p(zT ) = k2

[1 + λ1

(z3T − 3z)

6+ λ2

(z4T − 6z2 + 3)

24

]2

φ(zT ), (15)

where k2 is a normalizing constant and φ(·) represents the normal density. The term in squarebrackets is a polynomial containing terms up to the fourth order, which, in turn, is squared.This latter transformation ensures that the probabilities are non-negative. This compares with theJarrow and Rudd model where the polynomial in brackets is not squared, thereby opening up thepossibility of negative probability estimates.

Using (15) and (5) in (1) gives the SNP currency option price

FSNPt = e−rtτt k2

∫ ∞

X

(ST − X)

[1 + λ1

(z3T − 3z)

6+ λ2

(z4T − 6z2 + 3)

24

]2

φ(zT )|J |dST ,

(16)

where J is defined by (6) and zT is defined implicitly by (2). The normal distribution is a specialcase of (15), occurring when

λ1 = λ2 = 0,

in which case (16) represents the GK price with time-varying volatility. Skewness is accommo-dated when λ1 = 0 and excess kurtosis when λ2 = 0.

3. Estimation procedure

In this section a statistical model is developed whereby observed option prices are used toestimate the parameters of the theoretical option pricing model. More formally, the relationshipbetween Cj,t, the market price of the jth call option contract at time t, and Fj,t, the theoretical priceof the same option contract written at time t, is given by

Cj,t = Fj,t + ωej,t, (17)

where ej,t represents the pricing error with standard deviation ω. Following the approach of Engleand Mustafa (1992) and Sabbatini and Linton (1998), amongst others, the pricing error ej,t, isassumed to be an identically and independently distributed (iid) standardized normal randomvariable; see also the discussion in Renault (1997) and Clement et al. (2000).5 The theoreticaloption price is written as

Fj,t = F (St, Xj,t, τt, rt, it ; Ω), (18)

5 More general specifications of the pricing error in (17) could be adopted. For example, Lim et al. (2005) allow ω tovary across the moneyness spectrum of option contracts. Bates (1996, 2000) and Martin et al. (2005) adopt more generaldistributional structures for ej,t that incorporate both autocorrelation and heteroskedasticity, with structural shifts acrossmoneyness groups also accommodated in Martin et al. Bakshi et al. (1997) adopt an alternative approach by defining thestatistical model in terms of hedging errors.


where Ω is the vector of unknown parameters that characterize the returns distribution andthe volatility specification. In the special case of the GK option pricing model, for example,Ω = σ = exp(β0). Equation (17) may thus be viewed as a non-linear regression equation, withparameter vector, Ω.

The unknown parameters of the model can be estimated by maximum likelihood. The logarithmof the likelihood function is

lnL = −N

2ln(2πω2) − 1

2

∑j,t

(Cj,t − Fj,t

ω

)2

, (19)

where N is the number of observations in the panel of option prices. This function is maximizedwith respect to ω and Ω, using the GAUSS procedure MAXLIK. In maximizing the likelihood,ω is concentrated out of the likelihood. The numerical integration procedure for computing thetheoretical option price Fj,t for the various models, is based on the GAUSS procedure INTQUAD1.The accuracy of the integration procedure is ensured by checking that numerically and analyticallyderived GK prices yield parameter estimates that are equivalent to at least four decimal points. Thenumerical calculation of the non-GK theoretical option prices is extremely fast, and competitivewith the calculation of the GK option price based on the cumulative normal distribution. Finally, thenormalizing constant for the GST and SNP distributions in (14) and (16), k1 and k2, respectively,are also evaluated numerically using INTQUAD1.

4. Application to US/UK currency options

4.1. Data

The data set used in the empirical application consists of end-of-day European currency calloptions for the UK pound written on the US dollar, taken from the Bloomberg data base (see Bollenet al., 2000 and Bollen and Rasiel, 2003, for recent empirical applications based on end-of-daycurrency option prices). The sample period begins on October 1st, 1997 and ends on June 16th,1998, a time period of 178 days. The data set is restricted to contracts that mature in September1998. The complete set of strike prices over the sample period range from X = 158 to 178. Thus,the data represent a panel data set where the cross-sectional units correspond to the strike prices,constituting N = 736 observations in total.6

The US and UK risk free interest rates are the end-of-day 3-month Treasury bill rates. Theserates are relatively stable over the sample period, deviating only slightly from their respectivesample means of 5 and 7%. The US/UK exchange rate is the end-of-day value. The time seriesproperties of the exchange rate over the sample period are shown in Fig. 1.

To allow for ex post out-of-sample tests of the predictive performance of the alternative models,prices of option contracts occurring on the last 5 days of the sample are excluded from the sampleused for estimation. This leaves 705 observations for estimating the model and 31 observationsfor the forecasting experiments. Thus, estimation of the unknown parameters is based on dataending June 9th, 1998 and the forecast days are the 10th, 11th, 12th, 15th and 16th of June 1998.The number of option contracts on each day in the forecast period is, respectively, 4, 5, 4, 8 and10.

6 The data sets used by Bollen et al. (2000) and Bollen and Rasiel (2003) are of a similar size to that used here.


Fig. 1. US/UK exchange rate, October 1st, 1997 to June 16th, 1998.

4.2. Risk-neutral probability estimates

Tables 1–3 provide the parameter estimates of option price models based on alternative risk-neutral distribution assumptions. The distributions are the normal distribution (Table 1), denotedas NORM, the symmetric generalized Student t distribution (Table 2) with θ1 = 0 in (14), denotedas SGST, the generalized Student t distribution (Table 2) with θ1 = 0 in (14), denoted as GST, and

Table 1Maximum likelihood estimates and goodness of fit statistics for the GK model and the normal option price model,NORM(1) to NORM(3), based on alternative specifications of (4) given in (20), with quasi maximum likelihood standarderrors in brackets

Parameter NORM(1) NORM(2) NORM(3) GK

β0 −3.012 (0.011) −3.010 (0.015) −2.454 (0.003) −2.387 (0.003)β1 0.367 (0.002) 0.374 (0.034) 0.430 (0.001) 0.000β2 1.554 (0.039) 1.550 (0.048) 0.000 0.000β3 −0.942 (0.030) −0.939 (0.036) 0.000 0.000β4 −0.044 (0.108) 0.000 0.000 0.000β5 6.604 (4.323) 0.000 0.000 0.000

ln L/N 0.590 0.588 −0.177 −0.356s2 0.018 0.018 0.083 0.119AICa −820.494 −820.728 253.771 504.341AICb −793.145 −802.495 262.887 508.900

A value of zero without a standard error implies that the parameter is set equal to that value.a AIC = −2ln L + 2k, where L is the likelihood and k is the number of estimated parameters.b SIC = −2ln L + ln(N)k, where L is the likelihood, N the sample size and k is the number of estimated parameters.

G.C

.Lim

etal./J.ofMulti.F

in.Manag.16

(2006)291–314

301

Table 2Maximum likelihood estimates and goodness of fit statistics for the symmetric GST option price model, SGST(1) to SGST(3), and the GST option price model, GST(1) toGST(3), based on alternative specifications of (4) given in (20), with quasi maximum likelihood standard errors in brackets

Parameter SGST(1) SGST(2) SGST(3) GST(1) GST(2) GST(3)

β0 −2.960 (0.014) −3.001 (0.012) −2.451 (0.003) −2.959 (0.014) −2.977 (0.015) −2.618 (0.011)β1 0.307 (0.007) 0.332 (0.007) 0.414 (0.001) 0.306 (0.007) 0.191 (1.297) 0.520 (0.001)β2 1.606 (0.042) 1.566 (0.040) 0.000 1.607 (0.042) 1.612 (0.322) 0.000β3 −0.981 (0.033) −0.949 (0.031) 0.000 −0.981 (0.033) −0.971 (0.140) 0.000β4 −0.761 (0.212) 0.000 0.000 −0.748 (0.199) 0.000 0.000β5 −7.448 (3.966) 0.000 0.000 −7.734 (3.826) 0.000 0.000γ = √

ν 0.351 (0.086) 1.771 (0.613) 3.639 (0.067) 0.348 (0.080) 0.662 (0.440) 0.954 (0.063)θ1 0.000 0.000 0.000 −0.014 (0.037) −0.680 (0.574) −7.161 (0.451)

ln L/N 0.598 0.589 −0.169 0.598 0.593 0.562s2 0.018 0.018 0.082 0.018 0.018 0.019AICa −829.444 −821.160 245.380 −827.447 −824.438 −785.016SICb −797.537 −798.369 259.055 −790.981 −797.089 −766.783



Table 3Maximum likelihood estimates and goodness of fit statistics for the SNP option price model, SNP(1) to SNP(3), based onalternative specifications of (4) given in (20), with quasi maximum likelihood standard errors in brackets

Parameter SNP(1) SNP(2) SNP(3)

β0 −2.759 (0.045) −2.652 (0.020) −2.623 (0.006)β1 0.504 (0.005) 0.522 (0.004) 0.523 (0.001)β2 0.684 (0.196) 0.202 (0.114) 0.000β3 −0.422 (0.134) −0.196 (0.090) 0.000β4 0.719 (0.145) 0.000 0.000β5 −1.433 (9.143) 0.000 0.000λ1 −0.258 (0.010) −0.230 (0.008) −0.246 (0.004)λ2 0.222 (0.012) 0.193 (0.010) 0.215 (0.006)

ln L/N 0.579 0.556 0.551s2 0.018 0.019 0.019AICa −800.408 −772.094 −769.141SICb −763.943 −744.745 −750.908


the semi-non-parametric distribution (Table 3), denoted as SNP. In estimating the various optionprice models, three volatility specifications based on (4) are entertained,

Model 1 : βi = 0, i = 0, 1, 2, 3, 4, 5

Model 2 : βi = 0, i = 0, 1, 2, 3; β4 = β5 = 0

Model 3 : βi = 0, i = 0, 1; β2 = β3 = β4 = β5 = 0.

(20)

Model 1 represents the unconstrained model, as the volatility specification includes both matu-rity and moneyness factors. In Model 2, the moneyness factors are excluded, whereas in Model3 both moneyness and maturity factors are excluded. This last specification is also equivalentto the RE volatility specification in (3). For the set of volatility specifications where the distri-bution of the standardized returns is normal, the respective models are denoted as NORM(1),NORM(2) and NORM(3). Analogous acronyms are used for the SGST, GST and SNP specifi-cations. The GK model results given in Table 1 are based on a constant volatility model, whichis obtained by imposing the restrictions β1 = β2 = β3 = β4 = β5 = 0, on the volatility specificationin (4).

At the bottom of each table various measures of fit are reported, namely the mean log-likelihood,ln L/N, the residual variance, s2, and the AIC and SIC statistics. The residual variance, s2, is definedas

s2 =∑

j,t(Cj,t − Fj,t)2

N, (21)

where Cj,t and Fj,t are, respectively, the observed and estimated theoretical values of the jthcall option price at time t, with Fj,t evaluated at the maximum likelihood estimates of theparameters.

The important features of the empirical results presented in Tables 1–3 can be summarizedas follows. First, for all risk-neutral distributions volatility is found to be time-varying, withsignificant contributions from most of the terms in (4) occurring for all models. Most notable is


the impact on volatility of the future return over the life of the option, and the maturity effect terms.The impact of moneyness on volatility is slightly less marked overall, despite some significantestimates of β4 and β5.

Second, there is evidence that non-normal features in currency returns are priced in theUS/UK currency options. In particular, all estimates in Table 2 of the parameter that controlskurtosis in the SGST and GST models, namely γ = √

ν, are significant, apart from that forthe GST(2) model. All estimates of λ2 in Table 3, for the SNP models, are also significant.The evidence for skewness is slightly less marked in the case of the GST specifications, withestimates of the parameter θ1 being significant only for the GST(3) model. On the other hand,estimates of the skewness parameter for the SNP models, λ1, are significant for all versions of thatmodel.

Support for the importance of excess kurtosis in the risk-neutralised distributions is providedby the goodness of fit measures reported in Tables 1–3, with all such measures for the SGSTmodels being almost uniformly superior to the corresponding statistics for the NORM models.For the skewed GST(3) specification, the goodness of fit statistics are clearly superior to thoseassociated with its symmetric counterpart, SGST(3). For the other two pairs of GST and SGSTmodels, however, the fit statistics are very similar, with the symmetric versions actually providinga slightly better fit. A comparison of the comparable statistics in Tables 2 and 3 highlights the factthat the GST specification is superior to the SNP specification in terms of model fit, suggestingthat the non-normalities in the data are better captured using the GST family of distributions.Finally, the GK model clearly performs much worse than all other models considered, in termsof all fit measures, providing further evidence against the twin assumptions of normal currencyreturns and constant volatility.

More direct evidence of the non-normal properties of the estimated risk-neutralised distri-butions is provided in Table 4. This table provides both skewness and kurtosis statistics for all

Table 4Third and fourth order moments of the risk-neutral and empirical distributions of returns

Model Skewness Kurtosis

Statistic t-Statistic p-Value Statistic t-Statistic p-Value

NORM(1) 0.000 n.a. n.a. 3.000 n.a. n.a.NORM(2) 0.000 n.a. n.a. 3.000 n.a. n.a.NORM(3) 0.000 n.a. n.a. 3.000 n.a. n.a.

SGST(1) 0.000 n.a. n.a. 4.498 5.493 0.000SGST(2) 0.000 n.a. n.a. 3.326 2.148 0.032SGST(3) 0.000 n.a. n.a. 3.103 30.117 0.000

GST(1) −0.002 −0.359 0.720 4.505 5.810 0.000GST(2) −0.039 −0.414 0.679 3.952 0.344 0.731GST(3) −0.446 −20.004 0.000 3.290 11.576 0.000

SNP(1) −0.424 −120.939 0.000 3.333 21.779 0.000SNP(2) −0.407 −62.037 0.000 3.363 128.004 0.000SNP(3) −0.418 −193.369 0.000 3.354 87.454 0.000

Empirical 0.348 1.889 0.059 3.583 1.583 0.113

The statistics are calculated by evaluating (22) using a numerical integration procedure. The t-statistics are based on nullhypotheses of 0.0 and 3.0 when testing for skewness and kurtosis, respectively. The standard errors are computed usingthe delta method with the gradients computed using a numerical derivative with a step length of h = 0.00001.


estimated risk-neutral distributions computed, respectively, as

S =∫

z3T p(zT )dzT

K =∫

z4T p(zT )dzT ,

(22)

where p(zT) is the risk-neutral distribution based on either the NORM, SGST, GST or SNP modeldistributions.7 In the case of the normal distribution models, NORM(1), NORM(2) and NORM(3),the skewness and kurtosis values are simply the population values of 0.0 and 3.0, respectively.In the case of the symmetric GST distributions, SGST(1), SGST(2) and SGST(3), the skewnessvalues are set equal to their population value of 0.0. The t-statistics are based on standard errorscomputed using the delta method.

The risk-neutralised skewness estimates in Table 4 are consistent across the GST and SNPmodels, with the point estimates showing evidence of negative skewness. These estimatesare statistically significant in the case of all three SNP models and the GST(3) model, butnot for the GST(1) and GST(2) models, as is consistent with the significance results for θ1,reported in Table 2. The negative skewness feature of the risk-neutralised distributions con-trasts with the empirical skewness estimate reported in the last row of Table 4, which is positiveand statistically significant at the 5% level. This suggests that the option market has factoredin a higher probability of large falls in the exchange rate than is consistent with historicalexchange rate data (see Bates, 2000, for a discussion of this point in the context of equityoptions).

All risk-neutralised kurtosis estimates reported in Table 4 are in excess of 3.0, the kurto-sis value of the normal distribution. The t-statistics reported provide a test of the hypothesisthat the kurtosis estimate is not significantly different from 3.0. The hypothesis is rejected atthe 5% significance level for all models with the exception of GST(2), where the p-value is0.731. Again, this result is consistent with the estimation results reported in Table 2. The risk-neutralised kurtosis estimates range from 3.103 to 4.498, a range that includes the empiricalkurtosis estimate of 3.583, reported in the last row of Table 4. Interestingly, the empirical kur-tosis estimate is found to be statistically insignificant, in contrast to the risk-neutralised kurtosisestimates.

Comparisons of the GK risk-neutral distribution with the NORM(1), SGST(1), GST(1) andSNP(1) risk-neutral distributions at the time the option matures, are given in Fig. 2. The distri-butions are based on St = 165, rt = 0.05, it = 0.07 and τt = 0.5. The distribution of the NORM(1)model gives less weight to low values of ST than does the GK distribution, but slightly moreweight to values of ST around the centre of the distribution, at approximately ST = 165. For valuesof ST in the upper tails of the distributions, the probabilities of the GK and NORM(1) distributionsare very similar. This implies that the GK risk-neutral distribution predicts future falls in theexchange rate with higher probability than the NORM(1) distribution, whilst their predictionsof future increases in the exchange rate are comparable. The tail behaviour, upper and lower, ofthe SGST(1) and GST(1) models is similar to that of the GK model. However, the GK modelassigns much less weight to values of ST at the centre of the distribution than do the SGST(1)and GST(1) models. The distributions of the latter two models are very similar, reflecting the

7 An alternative distribution-free measure of the risk-neutralised skewness and kurtosis statistics is adopted in Bakshiand Madan (2000) and Bakshi et al. (2003). However, this approach requires the use of both call and put options data.


Fig. 2. Comparison of the Garman-Kohlhagan (GK) risk-neutral distribution with the NORM(1), SGST(1), GST(1) andSNP(1) risk-neutral distributions.

insignificance of the estimate of θ1 for GST(1) in Table 2. A comparison of the distributionsof the SNP(1) and GK models reveals that the SNP(1) risk-neutral distribution attaches lessweight to the upper tail and greater weight to values of ST around ST = 165. That is, the SNP(1)model predicts future rises in the exchange rate with smaller probability than does the GKmodel.

The behaviour of the risk-neutral distribution over the life of the option is demonstrated in Fig. 3for the case of the GST(1) distribution. As before the distributions are based on St = 165, rt = 0.05and it = 0.07. The range of maturities is τt = 0.1, 0.2, . . ., 0.5. These distributions show that as theoption contract approaches maturity, τt = 0, the risk-neutral distribution becomes more peaked.


Fig. 3. Plot of the GST(1) risk-neutral probability distribution for τt = 0.1, 0.2, . . ., 0.5, maturities.

This property stems from the estimated volatility function which, according to the parameterestimates reported in Table 2, implies a decrease in volatility as the time to maturity declines, allother things equal.8

An estimate of the volatility in (4) can be computed at each point in time. To perform thiscalculation it is necessary to choose a value for the exchange rate at the time of maturity, ST.Setting ST = 165.260, yields a volatility estimate of 9.882% at the start of the sample period, forat-the-money options and using the GST(1) parameter estimates. By the end of the estimationperiod, the volatility estimate falls to 7.487%. Performing the calculation at maturity, the volatilityestimate falls marginally further to 7.303%. As a check on these calculations, a GARCH(1,1)model is estimated using daily currency returns over the sample period. The estimatedmodel is

100(ln St − ln St−1) = 0.0241 + et

σ2t = 0.0183 + 0.0409e2

t−1 + 0.8745σ2t−1.

This yields a long-run value of the squared volatility of 0.0183/(1 − 0.0409 − 0.8745) = 0.2163.The long-run annualized volatility estimate is then 100

√(252)(0.2163) = 7.354%, which is very

similar to the implied risk-neutralised volatility estimate at maturity obtained from the GST(1)model. As a further comparison, the estimate of the annualized historical volatility is given by100 × S.D.

√252 = 7.220%, where S.D. is the standard deviation of currency returns for the

period.

8 Given τt < 1, and the magnitude of the relevant parameter estimates in Table 2, the positive impact of τt on σT|tdominates the negative impact of τ2

t on σT|t.


4.3. Forecasting

The model comparisons presented above are all based on within-sample statistical properties.In this section, the relative out-of-sample forecasting performance of the alternative models isinvestigated. Based on the parameter estimates reported above, predicted option prices are pro-duced for each contract in the five out-of-sample days. Poor forecasting properties will reflectevidence of misspecification and/or that the parameter estimates are not stable over the forecastperiod.

The forecast errors are assessed using two statistics. The first statistic reported is the root meansquared error (RMSE)

RMSE =√√√√ 1

Nf

∑j

fej,t, (23)

where fej,t denotes the forecast error for the jth contract at time t based on a particular model andNf denotes the number of observations in the forecast period. The RMSE of the various modelsare reported in Table 5. The forecast horizon consists of 5 days beginning June 10th (Wednesday)and ending June 16th (Tuesday), 1998. The RMSE is also reported for all five out-of-sample daystaken together in the last column.

The results of the forecasting tests reported in Table 5 show that the GK model yields theworst forecasts overall, as it produces by far the largest RMSE on the first four of the fiveforecast days. Allowing for some form of time variation in volatility (NORM) yields improve-ments in forecasting accuracy relative to the GK model on the 4 days for which the GK modelis inferior. Allowing for fatness in the tails of the conditional distribution, but no skewness(SGST), results in further improvements in forecast performance, although the gains are onlymarginal. Extending the returns distribution to allow for skewness yields little extra improve-ment, except in the case of GST(3), where the gains in forecast accuracy (over the corresponding

Table 5Forecasting performance of competing models across various daysa based on RMSE using (23), beginning June 10th(Wednesday) and ending June 16th (Tuesday), 1998

Model Day 1 (4) Day 2 (5) Day 3 (4) Day 4 (8) Day 5 (10) All 5 days (31)

GK 0.893 0.995 0.741 0.705 0.100 0.682

NORM(1) 0.269 0.377 0.108 0.067 0.532 0.356NORM(2) 0.270 0.376 0.109 0.067 0.532 0.355NORM(3) 0.775 0.878 0.623 0.586 0.128 0.588

SGST(1) 0.266 0.385 0.107 0.065 0.524 0.352SGST(2) 0.268 0.374 0.107 0.063 0.530 0.353SGST(3) 0.771 0.875 0.620 0.583 0.132 0.586

GST(1) 0.266 0.386 0.107 0.065 0.524 0.353GST(2) 0.267 0.384 0.107 0.068 0.530 0.355GST(3) 0.374 0.512 0.223 0.195 0.457 0.380

SNP(1) 0.327 0.466 0.175 0.147 0.498 0.372SNP(2) 0.358 0.502 0.206 0.179 0.473 0.378SNP(3) 0.343 0.491 0.192 0.167 0.487 0.378

a The number of observations in each forecasting period is in parentheses.


symmetric SGST(3) model) are quite substantial for the first 4 days. Finally, the SNP modelyields RMSE’s that are reasonably invariant to the volatility specification. Overall this modeldoes not perform as well as the best forecasting models within the NORM, SGST and GSTspecifications.

The second statistic used to gauge the relative forecasting performance of the alternative modelsis the Diebold and Mariano (1995) (DM) statistic. This statistic is used to assess the significanceof the difference between the forecast errors of the non-GK models and the GK model. Definingdj,t as the difference between the forecast error for the jth contract on day t of any particular modeland the forecast error from the GK model, the DM statistic is defined as

DM =√

Nf d

sd, (24)

where

d =∑

jdj,t

Nf

; sd =√∑

jd2j,t

Nf

.

Under the null hypothesis of no difference between the forecast errors, DM is asymptoticallydistributed as N(0,1).

The results of the Diebold–Mariano test are reported in Table 6. This table reports the p-valuesbased on a two-sided test using the asymptotic distribution. The results show that all of the modelsgenerate forecast errors that are statistically different at the 5% level from those associated withthe GK model. This result occurs for all of the five forecast days.

4.4. Implications for hedging

4.4.1. Delta hedgingThe empirical results highlight the importance of pricing the non-normalities in the currency

returns distribution and the time variations in the volatility process. The implications of these

Table 6Forecasting performance of competing models across various daysa based on the p-values of the Diebold–Mariano testusing (24), beginning June 10th (Wednesday) and ending June 16th (Tuesday), 1998

Model Day 1 (4) Day 2 (5) Day 3 (4) Day 4 (8) Day 5 (10) All 5 days (31)

NORM(1) 0.047 0.026 0.047 0.005 0.004 0.000NORM(2) 0.047 0.026 0.047 0.005 0.004 0.000NORM(3) 0.047 0.026 0.047 0.005 0.004 0.000

SGST(1) 0.048 0.027 0.048 0.005 0.005 0.000SGST(2) 0.047 0.026 0.047 0.005 0.004 0.000SGST(3) 0.047 0.026 0.048 0.005 0.005 0.000

GST(1) 0.048 0.027 0.048 0.005 0.005 0.000GST(2) 0.047 0.026 0.048 0.005 0.005 0.000GST(3) 0.046 0.026 0.046 0.005 0.004 0.000

SNP(1) 0.046 0.026 0.046 0.005 0.004 0.000SNP(2) 0.046 0.027 0.046 0.005 0.004 0.000SNP(3) 0.046 0.027 0.046 0.005 0.004 0.000

a The number of observations in each forecasting period is in parentheses.


departures from the GK model are now investigated in the context of constructing hedged portfo-lios. In the case of a delta-hedged portfolio, the expression of the delta is given as the derivativeof the price of the call option Ft, with respect to the exchange rate St. Under the assumptions ofconstant volatility and normal returns, an analytical expression for the delta for the GK model isgiven by

δGK = dFt

dSt

= e−itτtΦ(d1), (25)

where d1 is defined in (12). In the case of the option price models based on non-normal distributionsand/or non-constant volatility, the delta is computed using a numerical forward difference schemewith a step length of h = 0.001; that is, the option is evaluated at St and St + h, with the differencedivided by h.

The effects of different distributional assumptions on the delta hedge ratio are highlighted inTable 7. The calculations are based on a spot exchange rate equal to St = 165, with the domesticand foreign interest rates set at 0.05 and 0.07%, respectively. The contract is for a 3 monthoption, τt = 0.25, with strike prices ranging from X = 155 to 175 in steps of 5, to allow for a broadmoneyness spectrum. These values are based on the data set used in the empirical illustration.The parameter estimates used are those reported in Tables 1–3.

A comparison of the deltas across the alternative models reveals that the GK model hasuniformly lower delta values for the in-the-money contracts, X = 155 and 160. For the at-the-money option, X = 165, there is widespread agreement amongst all models, whilst for theout-of-the-money contracts, X = 170 and 175, the GK model produces uniformly higher deltavalues than the other models. That is, use of the delta associated with the GK model pro-duces too few units of foreign exchange in the hedged portfolio, for in-the-money options,relative to the number of units produced by the models that cater for various of the empir-ical features of exchange rate returns. The opposite situation arises in the case of the out-of-the-money options. These results suggest that by not pricing higher-order moments inexchange rate returns the portfolio is exposed, at least for contracts that are away from themoney.

Table 7Deltas of competing models across various strike prices: St = 165, rt = 0.05, it = 0.07, τt = 0.25

Model X = 155 X = 160 X = 165 X = 170 X = 175

GK 0.883 0.708 0.458 0.227 0.084

NORM(1) 0.942 0.772 0.442 0.155 0.035NORM(2) 0.943 0.769 0.441 0.159 0.035NORM(3) 0.903 0.723 0.455 0.214 0.076

SGST(1) 0.926 0.796 0.432 0.149 0.045SGST(2) 0.940 0.773 0.439 0.155 0.037SGST(3) 0.904 0.725 0.454 0.213 0.075

GST(1) 0.928 0.796 0.433 0.148 0.045GST(2) 0.930 0.780 0.444 0.145 0.040GST(3) 0.924 0.768 0.474 0.171 0.027

SNP(1) 0.960 0.747 0.424 0.190 0.071SNP(2) 0.926 0.778 0.472 0.169 0.031SNP(3) 0.927 0.781 0.473 0.164 0.028


4.4.2. Risk managementThe implications of constructing delta hedges in the presence of non-normalities in currency

returns are explored further by performing the following dynamic hedging experiment (see alsoBakshi et al., 1997). Consider selling European currency call options of 1000 units of foreignexchange at a strike price X, maturing in 3 months, τt = 0.25.9 The holder of the contract has theright to purchase 1000 units of foreign currency at a price of 1000 × 165 in domestic currency. Letthe domestic and foreign interest rates over the life of the option be fixed at rt = 0.05 and it = 0.07,respectively, and the volatility be equal to a constant value of 0.07. The distribution of zT in (13)is assumed to be a GST distribution with γ = 1 and θ1 = −1. Eq. (2) is then used to simulate theexchange rate over the life of the contract with the time interval equal to 1 day, with the initialspot rate set at St = 165. The chosen specifications reflect features of the empirical results reportedin Section 4.2.

Consider setting up a portfolio to minimize the exposure to the call option contract by using adelta hedge to determine the number of units of foreign assets to buy at the prevailing exchangerate. The cost of the initial investment in the domestic currency is

I0 = QCFt + QSSt,

where QC = −1000 represents the number of call options, Ft the price of a call option, QS = δ1000the total number of units of foreign exchange purchased based on the delta hedge parameter δ andSt is the spot exchange rate at time t. The term QCFt is the total value of the option contract andQSSt is the value of the holdings of foreign assets, both expressed in the domestic currency.

The portfolio is assumed to be rebalanced daily over the full life of the option, which results ineither foreign assets being sold or purchased, depending on the number of assets that are neededin the portfolio as determined by the delta hedge. In the case where assets are sold (bought) themoney is invested (borrowed) at the risk free rate of interest r. In computing the hedged portfolios,the deltas are based on three models: GK, SGST with γ = 1, and GST with γ = 1 and θ1 = −1. Foreach model, the volatility is assumed to be constant at 0.07.

The results of the dynamic hedging experiment are given in Table 8. Five separate contractsare considered with strike prices of 155–175 in steps of 5. For each experiment the table providesthe initial investment of each contract (I0), the value of the portfolio at the time the option matures(V), and the value of the investment (I) if I0 is invested at the risk free rate of interest (r = 0.05)over the life of the option,

I = I0er×0.25.

A measure of the performance of the portfolio is given by the percentage error between thevalue of the portfolio and investing in bonds,

Error = 100V − I

I.

For a perfectly hedged portfolio the return on the portfolio would equal r, the domestic risk-freerate of interest, so V = I, and hence Error = 0. An alternative measure of the performance of theportfolio given in Table 8 is the annualized return on the portfolio

rportfolio = ln

(P

I0

)(1

0.25

).

9 In the case of call options for the British pound, the sizes of contracts in practice are 31,250 pounds (Hull, 2000).


Table 8Dynamic hedging experiments over a 90 day period across various strike prices, with St = 165, r = 0.05, i = 0.07, τ = 0.25

Strike (X) Hedgingmodel

I0 ($) V ($) I ($) Error (%) rportfolio

(%p.a.)rbonds

(%p.a.)

155 GK 1450.96 1445.96 1469.21 −1.61 −1.89 5.00SGST 1450.96 1445.57 1469.21 −1.61 −1.96 5.00GST 1447.66 1441.74 1465.87 −1.65 −2.24 5.00

160 GK 1206.88 1199.32 1222.06 −1.86 −2.69 5.00SGST 1233.28 1225.64 1248.79 −1.85 −2.66 5.00GST 1231.63 1223.12 1247.12 −1.92 −3.00 5.00

165 GK 710.44 707.02 719.37 −1.72 −2.03 5.00SGST 700.54 697.17 709.35 −1.72 −2.04 5.00GST 712.09 709.66 721.14 −1.58 −1.52 5.00

170 GK 260.80 259.37 264.08 −1.78 −2.35 5.00SGST 241.00 239.50 244.03 −1.86 −2.72 5.00GST 236.05 234.43 239.02 −1.92 −3.08 5.00

175 GK 56.96 56.81 57.68 −1.51 −1.31 5.00SGST 60.26 60.14 61.02 −1.44 −1.04 5.00GST 56.96 56.77 57.68 −1.57 −1.71 5.00

The volatility is assumed to be constant and equal to 0.07. Currency returns are assumed to be drawn from a GST distributionwith parameters γ = 1 and θ1 = −1 in (13). Rebalancing is undertaken daily over the life of the option. Reported valuesare based on mean estimates from 1000 Monte Carlo simulations. The initial value of the investment is given by I0, Vis the value of the hedged portfolio at the time of maturity, and I is the value of a portfolio at the time of maturity frominvesting in bonds.

Again, for a perfectly hedged portfolio, rportfolio will equal the risk free rate of interest r. Theexperiment is repeated 1000 times with the mean values of each of the quantities reported inTable 8.

The results in Table 8 demonstrate that the returns on all hedged portfolios are negative,indicating that the hedged portfolios all incur a loss relative to investing in bonds at the risk freerate (5%). For the at-the-money contract (X = 165), the GST hedged portfolio incurs the smallestloss of the three portfolios. For the remaining options the results are mixed, with the SGST modelperforming best in two cases and the GK model in the remaining two cases. The mixed nature ofthe results would appear to reflect the fact that the error that occurs due to the discrete nature ofthe rebalancing outweighs any gains to be had from using a hedged portfolio based on the truedata generating process, which is GST in this case.10

5. Conclusions

This paper has provided an alternative framework for pricing currency options in which thetwin assumptions of constant volatility and normal currency returns that underlie the commonlyused pricing model of Garman and Kohlhagen (1983), are relaxed. Time-variation in volatil-ity was achieved by allowing volatility to be a function of the future return over the life ofthe option, as well as additional factors that capture time to maturity and moneyness. The use

10 This point is confirmed by the fact that the results (not reported) are still mixed even when the true data generatingprocess is normal with constant variance.


of future returns contrasted with existing option price models that accommodate time-varyingvolatility using standard stochastic volatility or GARCH specifications. Non-normalities in therisk-neutral probability distribution were explicitly modelled using a range of non-normal con-ditional distributions that captured both skewness and fat tails. An important feature of theproposed modelling framework was that currency options could be priced in a computation-ally efficient manner, as the pricing of options entailed only the numerical evaluation of aone-dimensional real integral. This contrasted with models that require either the use of MonteCarlo simulation to evaluate option prices, or the use of numerical integration in the complexplane.

The proposed modelling framework was applied to pricing European currency call options onthe UK pound written on the US dollar over the period October 1st, 1997 to June 16th, 1998.The analysis was performed on a panel of call options with prices computed jointly on contractswithin days as well as across days. The key empirical results showed that the proposed option pricemodel resulted in large reductions in pricing errors and improvements in forecasting, comparedto the Garman and Kohlhagen model. These improvements arose from the adoption of a non-constant volatility specification and a conditional distribution that allowed for non-normalities.Of the alternative non-normal probability distributions used to model the underlying risk-neutralprobability distribution, the generalized Student t distribution performed the best overall in termsof capturing skewness and fatness in the tails of the underlying currency returns distribution. TheGarman and Kohlhagen currency option price model performed the worst in terms of all of thecriteria considered. Finally, the results of the hedging experiments showed that the generalizedmodel produced hedged portfolios that were indeed different from those based on the Garmanand Kohlhagen model. This implies, in turn, that pricing higher-order moments has importantimplications for constructing international hedged portfolios to minimize exposure to movementsin exchange rates.

Acknowledgement

We thank an anonymous referee for insightful comments and suggestions on a previous versionof the paper.

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pricing currency options in the presence of time-varying volatility and non-normalities

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