Pricing Currency Options in the Presence ofTime-Varying Volatility and Nonnormalities

G.C.Lim(a), G.M.Martin(b) and V.L.Martin(a)

October 10, 2003

Abstract

A new framework is developed for pricing currency options where the distributionof exchange rate returns exhibits time-varying volatility and nonnormalities. A forwardlooking volatility structure is adopted whereby volatility is expressed as a function ofcurrency returns over the life of the contract. Time-to-maturity e¤ects as well as levelse¤ects in volatility are also considered. Both skewness and fat-tails in currency returnsare priced using a range of nonnormal risk neutral probability distributions. A specialfeature of the approach is that an analytical solution for the option price is obtainedup to a one-dimensional integral in the real plane. This enables option prices to becomputed e¢ ciently and accurately, which is in contrast to existing methods that relyon Monte Carlo numerical procedures to price options in the presence of nonnormalreturns and time-varying moments. The proposed modelling framework is applied topricing European currency call options for the UK pound written on the US dollarusing daily data over the period October 1997 to September 1998. The results showthat the new approach yields, in general, more accurate within-sample �t and out-of-sample prediction of observed currency option prices, when compared with a rangeof competing option price models. The results also demonstrate that skewness hasimportant implications for constructing hedged portfolios and managing risk.

Key words: Option pricing; skewness; fat-tails; time-varying volatility; generalisedStudent t; lognormal mixture; semi-nonparametric.

JEL classi�cation: C13, G13

(a) Department of Economics, University of Melbourne, Victoria, Australia.

(b) Department of Econometrics and Business Statistics, Monash University, Victoria, Aus-tralia.

1

1 Introduction

One of the main challenges in devising empirical models of currency options is to specify

models that are rich enough to capture the empirical characteristics commonly observed in

currency returns, whilst being computationally simple to evaluate. Well-established empiri-

cal characteristics of currency returns distributions are excess kurtosis and skewness, as well

as time varying moments.1 None of these empirical characteristics are priced in the Garman

and Kohlhagen (1983) model of currency options, which assumes normal currency returns

and constant volatility. However, as the Garman and Kohlhagen (GK) model often misprices

currency options across the moneyness spectrum, more complex option price models have

been developed to take into account the observed features of currency returns.2 Some recent

examples are: Melino and Turnbull (1990), Heston (1993) and Guo (1998), which allow for

stochastic volatility; Bates (1996), which includes both jump processes and stochastic volatil-

ity; Lim, Lye, Martin and Martin (1998), which allows for GARCH time-varying volatility

and nonnormal currency returns; and Bollen, Gray and Whaley (2000), which adopts a

Markovian switching framework. For a recent analysis of currency options that compares

the empirical performance of some of the alternative time-varying moment models that have

been proposed, 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 dis-

tribution of the exchange rate at the time the option contract matures, with the parameters

of the distribution representing the risk-neutralised versions of their empirical analogues.

To allow for time-varying volatility, a volatility structure based on the work of Rosenberg

and Engle (1997) and Rosenberg (1998) is adopted.3 This structure represents forward-

looking behaviour, whereby the volatility is speci�ed as 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 of lagged currency returns.

The Rosenberg and Engle volatility speci�cation is also extended here to allow for additional

determinants to capture the time series patterns commonly observed in implied volatilities.

1For a review of the empirical evidence pertaining to currency returns, see de Vries (1994), and morerecently, Bai, Russell and Tiao (2003) and Jondeau and Rockinger (2003).

2One manifestation of the mispricing of the GK model is the occurrence of implied volatility smiles; seeMelino and Turnbull (1990), Hull (2000) and Bollen and Rasiel (2003).

3See Lim, Martin and Martin (2003) for a recent application of this pricing framework to equity options.

2

The choice of the determinants is in�uenced by the work of Dumas, Fleming and Whaley

(1998), who allow for maturity e¤ects, and Fung and Hsieh (1991), who investigate levels

e¤ects of the underlying asset on the implied volatility of currency option prices.4

An important feature of the proposed modelling framework is that the currency option

price is expressed as a one-dimensional integral in the real plane. In the special case of normal

currency returns and constant volatility, the integral is standard, being based on the log-

normal distribution. For the more general case of time-varying volatility and nonnormalities,

the integral is still one-dimensional, but needs to be computed numerically on the real plane.

This is in contrast to models that rely on Monte Carlo methods to price options, which can

be computationally demanding (Hafner and Herwartz, 2001, Bauwens and Lubrano, 2002,

Lehar, Scheicher and Schittenkopf, 2002) and potentially less accurate unless a very large

number of simulations are used (Hull, 2000). Even in the case of the stochastic volatility

model proposed by Heston (1993) and the GARCH time-varying volatility model proposed

by Heston and Nandi (2000), the computational requirements are still more demanding than

the pricing framework proposed here, as integration needs to take place over the complex

plane. Moreover, these models still assume normal returns, and thus lack the �exibility of

the modelling framework developed here.5

In specifying the risk neutral distribution three candidates are used to capture the non-

normalities in the conditional distribution of currency returns: the generalised Student t

distribution (GST hereafter) of Lye and Martin (1993, 1994), the lognormal mixture distrib-

ution of Melick and Thomas (1997) and the semi-nonparametric distribution (SNP hereafter)

of Gallant and Tauchen (1989). The use of the GST distribution is motivated, in part, by

the simulation experiments in Lim, Lye, Martin and Martin (1998), which demonstrate that

this distribution has su¢ cient �exibility to price currency options.6 The use of the lognormal

mixture distribution has the advantage that the option price is expressed in terms of known

one-dimensional integrals, although the computational gains in terms of speed and accuracy

relative to using the GST distribution are, at best, marginal. The SNP distribution has

recently been used to model the dynamics of asset markets by Chernov, Gallant, Ghysels

4Other potential determinants of the implied volaltility are intra-day e¤ects and policy announcemente¤ects which are considered by Kim and Kim (2003).

5Bollen and Rasiel (2003) use a lattice approach to price currency options for alternative time-varyingmoment speci�cations, but still assume conditional normality for the currency returns distribution.

6In these experiments a GARCH volatility model is adopted, resulting in the need to price options usingMonte Carlo methods.

3

and Tauchen (2003) and represents an alternative to the GST distribution for capturing

nonnormalities in returns data. This distribution was recently used in the context of pricing

equity options by Lim, Martin and Martin (2003). The SNP option pricing model related

to the option pricing model proposed by Jarrow and Rudd (1980) and applied by Corrado

and Su (1997) and Capelle-Blancard, Jurczenko and Maillet (2001), amongst others. How-

ever, in contrast to the Jarrow and Rudd model, the semi-parametric pricing method based

on the Gallant and Tauchen distribution ensures non-negative estimates of the risk-neutral

probabilities and, hence, is a valid model for the pricing of currency options; see Lim, Martin

and Martin (2003) for more on this point.7

The modelling framework is applied to the pricing of European currency call options for

the UK pound written on the US dollar over the period October 1997 to June 1998. A range

of within-sample and out-of-sample experiments are performed, in order both to compare

the di¤erent distributional speci�cations and to assess the gains to be had by generalising

beyond the GKmodel for currency options. The results show strong evidence of time-varying

volatility over the sample period as well as signi�cant nonnormalities in the distribution of

currency returns, thereby validating the more general pricing approach. In addition, they

demonstrate that the option price model in general performs better when the risk-neutralised

distribution is speci�ed as the GST distribution rather than as either the lognormal mixture

or semi-nonparametric distribution. An important feature of the results is the identi�cation

of skewness having been priced in the UK/US currency options over the sample period

investigated. These implications are investigated further in the context of devising dynamic

hedging strategies to minimise the portfolio risk of exposure to exchange rate movements.

The rest of the paper proceeds as follows. Section 2 presents the option pricing frame-

work that combines time-varying volatility and a �exible risk neutral probability distribution

speci�cation to capture nonnormalities in currency returns. A maximum likelihood proce-

dure is proposed in Section 3 to estimate the parameters of the model. The empirical

performance of the competing models is investigated in Section 4 using both within-sample

and out-of-sample criteria. The implications of nonnormalities and time-varying volatility

7Of course a range of other models could be used to capture the nonnormalities in currency returns,such as Binomial trees (Jackwerth and Rubinstein, 1996; and Dennis, 2001); Markovian switching (Bollen,Gray and Whaley, 2000); stochastic volatility with jumps (Bates, 1996); non-parametric kernel methods(Ait-Sahalia, 1996, Ait-Sahalia and Lo, 1998, and Ghysels, Patilea, Renault and Torres, 1998); and thegeneralised Student t distribution investigated by Jondeau and Rockinger (2003). These models are notpursued in the present paper.

4

for constructing hedged portfolios and 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

that has su¢ cient parametric �exibility to capture the empirical characteristics of currency

returns. A special feature of this parametric model is that it nests a number of pricing

models, including the GK option pricing model, which is equivalent to the Black and Scholes

(1973) model for pricing European call options 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

that all parameters speci�ed and estimated are risk-neutralised quantities. Associated with

these risk-neutralised parameters are the parameters of the underlying empirical distribution

of currency returns which, in turn, can be used to extract an estimate of the risk preferences

factored into the option prices. To this extent the approach is similar to the nonparametric

method adopted by Jarrow and Rudd (1982) and applied subsequently by Corrado and

Su (1997) and Capelle-Blancard, Jurczenko and Maillet (2001), amongst others, with the

risk-neutral distribution speci�ed as a Gram-Charlier distribution; see also Bollen 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 the domestic currency. De�ne rt and it as the respective domestic and foreign risk free

annualised interest rates at time t for maturity in n periods, at time t+n. Consider writing

a European call option at time t on St with strike price X; that matures at time t+ n. The

price of the currency option is; see Hull (2000)

F (St) = Et�e�rt� max (St+n �X; 0) jSt

�= e�rt�

1ZX

(St+n �X)g(St+njSt)dSt+n; (1)

where Et denotes the expectation formed at time t with respect to the risk-neutral distribu-

tion for the exchange rate at the time of maturity, g(St+njSt); and � = n=252 is the maturity

5

of the option contract expressed as a proportion of a year. In this paper, the risk-neutral

distribution for the continuously compounded return over the life of the option is assumed

to be represented by

ln

�St+nSt

�=

rt � it �

�2t+njt2

!� + � t+njt

p�z; (2)

where z is a zero mean, unit variance random variable that represents unanticipated move-

ments in exchange rates and that is uncorrelated with rt � it � �2t+njt=2; and � t+njt is theconditional volatility, for which the speci�cation is detailed in the following section. The

speci�cation of the distribution of z determines, in combination with the speci�cation of

� t+njt; the form of the risk neutral distribution used to price options.

2.1 Conditional Volatility

The two models most commonly adopted to capture time-variation in volatility when pricing

currency 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, Lye, Martin and Martin (1998), Hafner and Herwartz (2001)

and Bauwens and Lubrano (2002), amongst others. In both classes of models, the volatility

speci�cation is backward-looking as it is lagged returns that in�uence current volatility.

An alternative approach which is adopted here is to allow the volatility structure to be

forward-looking, whereby the volatility is speci�ed to be a function of the return over the

remaining life of the contract. This framework was originally proposed by Rosenberg and

Engle (1997) and applied by Rosenberg (1998) and Lim, Martin and Martin (2003).

The Rosenberg and Engle (RE) volatility speci�cation is given by

ln� t+njt = �0 + �1 ln (St+n=St) ;

which shows that the natural logarithm of volatility is a linear function of the currency

return over the life of the option contract, namely ln (St+n=St) : As the remaining time until

the contract matures decreases, St approaches St+n: On the day that the contract matures,

n = 0; the volatility reduces to � t+njt = exp (�0). 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 additional random error term. The randomness in volatility

6

arises from the randomness in the spot price at the time of maturity, St+n; with this future

price being unknown at time t. This means that the option market is complete, with there

being, as a consequence, no need to resort to additional equilibrium arguments in order to

price options as is the case with volatility speci�cations based on stochastic volatility.

The volatility speci�cation adopted in the present paper extends the RE speci�cation

above to include three additional terms, � t; � 2t and lnSt;

ln� t+njt = �0 + �1 ln (St+n=St) + �2� t + �3�2t + �4 lnSt: (3)

The terms � t and � 2t in (3) allow for deterministic time-variation whereby volatility is a

function of the time to maturity. This speci�cation is motivated by the work on volatility

functions by Dumas, Fleming and Whaley (1998) and is adopted in the empirical analysis

conducted below to help capture the observed time-variation in implied volatility over the

sample period investigated. The remaining term, lnSt; is included to allow for a levels e¤ect

in volatility, as motivated by the work of Fung and Hsieh (1991), who investigate the e¤ects

of the level of the underlying asset price on implied volatility in currency options markets.

2.2 The Risk Neutral Distribution

Given (2) and (3), the risk neutral probability distribution at the time of maturity, g(St+njSt);is given by

g(St+njSt) = jJ j p (z) ; (4)

where J is the Jacobian of the transformation from z to St+n; de�ned as

J =dz

dSt+n=

1 + �1�2t+njt� � �1

�ln (St+n=St)�

�rt � it �

�2t+njt2

��

�St+n� t+njt

p�

: (5)

As is clear from (5), the allowance for time-varying volatility in the option price formulation

is not associated with any additional computational complexity. The speci�cation in (3),

being a function of St+n; enables a closed form solution for the Jacobian in (5) to be produced.

This contrasts, 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 conditional normality. Moreover, the solution that is produced in the case of the Heston

model is analytical only up to two one-dimensional integrals in the complex plane, a situation

7

that contrasts with the single real integral whose evaluation is required in (1).8

The computational simplicity associated with the use of the time-varying volatility struc-

ture in (3) di¤ers from the computational burden associated with the use of a GARCH-type

speci�cation. 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 GARCHmodel

with a nonnormal conditional distribution, as in Hafner and Herwartz (2001) and Bauwens

and Lubrano (2002), amongst others, entails the use of computationally intensive Monte

Carlo simulation for the option price evaluation.

2.3 Alternative Distributional Forms for z

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

still true even when time-varying conditional moments are used (Bai, Russell and Tiao, 2003).

To capture these empirical features, z in (2) is assumed to have a nonnormal distribution.

Three types of nonnormal distributions are considered: the GST distribution of Lye and

Martin (1993, 1994), the lognormal mixture distribution of Melick and Thomas (1997) and

the semi-nonparametric distribution of Gallant and Tauchen (1989). Whilst there are many

other choices that could be contemplated for the risk neutral probability distribution, these

speci�cations all share the common advantage that they can provide a convenient nesting of

the lognormal distribution and, as a consequence, the GK currency option price model.

2.3.1 Generalised Student t

The general form of the GST distribution is given by (Lye and Martin, 1993)

p (z) = k�1�w exp [�1 tan�1��w + �wzp

�

�+ �2 ln

�� + (�w + �wz)

2�+

4Xj=1

�j+2 (�w + �wz)j ]; (6)

8Estimation of Heston-type models using observed option prices also has the additional computationalburden associated with the presence of the latent volatilities; see, for example, Guo (1998), Bates (2000) andChernov and Ghysels (2000).

8

where �w and �w are chosen to ensure that z is standardised to have zero mean and unit

variance, and k is the normalising constant, de�ned by

k =

Zp (z) dz: (7)

The moments of the distribution in (6) exist as long as the parameter on the highest even-

order term, namely �6 in (6); is negative. The power term,�� + (�w + �wz)

2��2 ; is a general-ization of the kernel of a Student t density and the parameters in that term, � and �2, along

with the parameters �4 and �6, control the degree of kurtosis in the distribution. The odd

moments of the distribution, including functions of them such as skewness, are controlled by

the parameters �1; �3 and �5:

The price of the currency option as de�ned by (1) to (7) nests a number of special cases.

Setting

�1 = �2 = �3 = �4 = 0; (8)

in (3) results in a constant volatility model with nonnormal currency returns. Further

imposing the restrictions

�4 = �0:5; and �j = 0;8j 6= 4; (9)

in (6) yields the GK model, as p (z) reduces to the standardised normal distribution and

g(St+njSt) in (4) becomes lognormal. In this special case of constant volatility and normalcurrency returns, the GK option price is

FGK = Se�i�N(d1)�Xe�r�N(d2); (10)

where

d1 =ln(S=X) +

�r � i+ �2

2

��

�p�

;

(11)

d2 =ln(S=X) +

�r � i� �2

2

��

�p�

:

In the empirical analysis conducted below, the form of the GST distribution speci�ed for

z is

p (z) = k�w exp [��1 + �

2

�ln�� + (�w + �wz)

2�+ � (�w + �wz)�0:25 (�w + �wz)

4 ]; (12)

9

where � controls skewness and � models the thickness in the tails of the distribution. The

speci�cation of the negative sign on the fourth order polynomial term ensures that all mo-

ments of the distribution exist, whilst the magnitude of 0:25 is arbitrary and serves to identify

the standardisation factor �w:9 A similar normalisation is adopted for the normal distrib-

ution whereby a value of �0:5 is chosen for the second order polynomial term to identify

�w:

Using (12) in (4), which, in turn, is substituted into (1), gives the GST currency option

price

FGST = e�rt�k

1ZX

(St+n �X)�w exp [��1 + �

2

�ln�� + (�w + �wz)

2�+� (�w + �wz)�0:25 (�w + �wz)

4 ] jJ j dSt+n; (13)

where J is de�ned by (5) and z is de�ned by rearranging (2).

2.3.2 Mixture of Lognormals

The second distribution investigated is the lognormal mixture distribution suggested by

Melick and Thomas (1997). The distribution of z in (2) is taken as a mixture of two normals

with mixing parameter � that assigns a non-zero probability of z being drawn from the

respective distributions. The option pricing model in this case is

FLMIX = �FGK��1; t+njt

�+ (1� �)FGK

��2; t+njt

�; (14)

where FGK��i; t+njt

�; i = 1; 2; is the GK price given by (10) but where volatility is allowed

to be time-varying, rather than being �xed as in the original analysis of Melick and Thomas.

Also in contrast to the original formulation of Melick and Thomas, risk neutrality in the

mean is imposed. The parameter, 0 � � � 1; is the mixing parameter which weights

the two subordinate lognormal distributions to generate skewed and fat-tailed risk neutral

9This generalisation of the Student t distribution circumvents the problems identi�ed by Duan (1999) inmodelling nonnormalities in option prices using the Student t distribution.

10

distributions.10 The corresponding subordinate volatilities are speci�ed as

�1; t+njt =�exp �1;0 + �1;1 ln (St+n=St) + �1;2� t + �1;3�

2t + �1;4 lnSt

��2; t+njt =

�exp �2;0 + �2;1 ln (St+n=St)

�: (15)

The volatility speci�cation associated with the �rst distribution contains the full set of

explanatory variables speci�ed in (3), whereas the second volatility speci�cation tallies with

the more restrictive Rosenberg-Engle formulation.11

2.3.3 Semi-Nonparametric

The form of the semi-nonparametric distribution adopted here is based on augmenting a

normal density for the standardised variable z in (2) by a polynomial that captures higher

order moments. Jarrow and Rudd (1982) were the �rst to adopt this approach, which has

more recently been implemented by Corrado and Su (1997) and Capelle-Blancard, Jurczenko

and Maillet (2001). As the adopted distribution is based on a Taylor series expansion around

the GK option price distribution, that is the log-normal distribution, Jarrow and Rudd

interpret this distribution as a local risk neutral distribution.

A problem that arises in using the Jarrow and Rudd (1982) speci�cation is that the

generated probabilities are not constrained to be non-negative. To impose non-negativity

on the underlying risk neutral probability distribution, an alternative speci�cation is used

here, based on the semi-nonparametric density of Gallant and Tauchen (1989). The form of

the standardised returns distribution is

p (z) = h

�1 + �1

(z3 � 3z)6

+ �2(z4 � 6z2 + 3)

24

�2n (z) ; (16)

where h is a normalising constant, and n (zT ) represents the normal density. The term in

square brackets 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

10One generalisation of the lognormal mixture model that was also adopted in the empirical applicationallowed the weighting parameter � in (14) to be time-varying. Time-variation in � was modelled using aMarkovian switching model following the approach of Bollen, Gray and Whaley (2000). This model did notyield results that were superior to the lognormal mixture model and, hence, the relevant results were notreported.11The initial speci�cation of the volatility of the second distribution included all explanatory variables.

However, this more general model experienced convergence problems in the estimation algorithm, suggest-ing that the additional parameterisation was redundant. As a consequence only the results based on thespeci�cation in (15) are reported.

11

compares with the Jarrow and Rudd model where the polynomial in brackets is not squared,

thereby opening up the possibility of negative probability estimates.

Using (16) and (4) in (1) gives the SNP currency option price

F SNP = e�rt�h

1ZX

(St+n �X)�1 + �1

(z3 � 3z)6

+�2(z4 � 6z2 + 3)

24

�2n (z) jJ j dSt+n; (17)

where J is de�ned by (5) and z is de�ned implicitly by (2).12 The normal distribution is a

special case of (16) occurring when

�1 = �2 = 0;

in which case (17) represents the GK price with time-varying volatility.

3 Estimation Procedure

In this section a statistical model is developed whereby observed option prices are used

to estimate the parameters of the theoretical option pricing model. More formally, the

relationship between Cj;t; the market price of the jth call option contract at time t, and Fj;t;

the theoretical price of the same option contract written at time t; is given by

Cj;t = Fj;t + !ej;t; (18)

where ej;t represents the pricing error with standard deviation !. Following the approach of

Engle and Mustafa (1992) and Sabbatini and Linton (1998), amongst others, the pricing error

ej;t; is assumed to be an iid standardised normal random variable; see also the discussion

in Renault (1997) and Clement, Gourieroux and Monfort (2000).13 The theoretical option

price is written as

Fj;t = F (St; Xj;t; � j; rt; it; ) ; (19)

where is the vector of unknown parameters that characterise the returns distribution and

the volatility speci�cation. In the special case of the GK option pricing model for example,

12The semi-nonparametric option price in (17) was introduced in Lim, Martin and Martin (2003) to pricenonnormalities in options on equities.13More general speci�cations of the pricing error in (18) could be adopted. For example, Lim, Martin

and Martin (2003) allow ! to vary across the moneyness spectrum of option contracts, while Bates (1996,2000) adopts a more general distributional structure for ej;t that incorporates both autocorrelation andheteroskedasticity. Bakshi, Cao and Chen (1997) adopt an alternative approach by de�ning the statisticalmodel in terms of hedging errors.

12

= f�0g. Equation (18) may thus be viewed as a nonlinear regression equation, withparameter vector, .

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

logarithm of the likelihood function is

lnL = �N2ln�2�!2

�� 12

Xj;t

�Cj;t � Fj;t

!

�2; (20)

where N is the number of observations in the panel of option prices. This function is

maximised with respect to ! and ; using the GAUSS procedure MAXLIK. In maximising

the likelihood, ! is concentrated out of the likelihood. The numerical integration procedure

for computing the theoretical 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 analytically derived GK prices yield parameter estimates that are

equivalent to at least four decimal points. The normalising constant for the GST and semi-

nonparametric distributions in (13) and (17), k and h respectively, are evaluated numerically

as well.14

4 Application to UK/US Currency Options

4.1 Data

The data set used in the empirical application consists of end-of-day European currency call

options for the UK pound written on the US dollar, taken from the Bloomberg data base.

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 September, 1998, so as to

focus on volatility structures across strike prices. 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.15

The US and UK risk free interest rates are the end-of-day 3-month Treasury bill rates.

These rates are relatively stable over the sample period, deviating only slightly from their

14The calculation of the non-GK theoretical option prices by numerical integration is extremely fast and,hence, competitive with the calculation of the GK option price based on the cumulative normal distribution.15Similar sized data sets are used by Bollen, Gray and Whaley (2000) and Bollen and Rasiel (2003), who

also use end-of-day currency option prices over a period of time.

13

Figure 1: US/BP exchange rate, October 1st, 1997 to June 16th, 1998.

respective sample means of 5% and 7%. The US/BP exchange rate is the end-of-day value.

The time series properties of the exchange rate over the sample period are shown in Figure

1.

To allow for some ex-post out-of-sample tests of the predictive performance of the alter-

native models, prices of option contracts occurring on the last �ve days of the sample are

excluded from the sample used for estimation. This leaves 705 observations for estimating

the model and 31 observations for the forecasting experiments. Thus estimation of the un-

known parameters is based on data ending June 9th, 1998. The forecast days are the 10th,

11th, 12th, 15th and 16th of June, 1998. The number of option contracts on each day are

respectively, 4,5,4,8 and 10.

4.2 Risk-Neutral Probability Estimates

In this section the results of estimating the parameters of various risk neutral probability

distributions are reported. To serve as a benchmark the results are presented initially for

the case of z in (2) being normal.

14

4.2.1 Normal

Table 1 reports the parameter estimates of various option price models based on the assump-

tion that z is normally distributed. Four volatility speci�cations are entertained. The �rst

three speci�cations, NORM(1) to NORM(3), allow for time-varying volatility, whereas the

last speci�cation is the GK speci�cation whereby volatility is assumed to be constant. At

the bottom of the table various measures of �t are reported, namely the mean log-likelihood,

lnL=N , the residual variance, s2, and the AIC and SIC statistics. The residual variance, s2,

is de�ned as

s2 =

Pj;t (Cj;t � Fj;t)

2

N; (21)

where Cj;t and Fj;t are respectively the actual and theoretical values of the jth call option

price at time t, with Fj;t evaluated at the maximum likelihood estimates of the parameters.

The results for NORM(1) show that all volatility parameter estimates of �i are statis-

tically signi�cant, with the exception of �4: The results for NORM(2) show that setting

�4 = 0 results in loss of goodness of �t as there is no change in s2, whereas there is an

improvement in the AIC and SIC statistics. Further restricting the volatility speci�cation

to exclude the maturity variables, by setting �2 = �3 = 0, the results associated with the

NORM(3) speci�cation show that this restriction does indeed result in a large loss of �t,

with the residual variance increasing from 0.018 to 0.083.

The results in the last column in Table 1 are achieved by setting �1 = �2 = �3 = �4 = 0:

These restrictions are seen to produce a large fall in explanatory power, with residual variance

increasing ten-fold relative to the residual variance of the NORM(1) and NORM(2) models,

in addition to a massive increase in both the AIC and SIC statistics. Of all the models

presented in Table 1, the GK model clearly performs the worst.

4.2.2 GST

Table 2 provides the parameter estimates of various option price models based on the GST-

based price in (13). The �rst three models reported are referred to as SGST(1) to SGST(3).

These models allow for various time-varying volatility structures and specify symmetric

GST currency returns distributions with fat-tails by setting � = 0 in (12). The next three

estimated models, GST(1) to GST(3), allow � to be unrestricted, thereby extending SGST(1)

to SGST(3) to accommodate skewness in the currency returns distribution.

15

A comparison of the estimates of the volatility parameters, �i, in Tables 1 and 2 respec-

tively, shows that the particular conditional returns distribution that is adopted has little

impact on the qualitative nature of the estimated volatility structure. Speci�cally, future

exchange rate returns and the time-to-maturity variables are signi�cant in all models, whilst

the level e¤ect of the exchange rate is insigni�cant for all models.

The point estimates of the skewness parameter, �, for models GST(1) to GST(3) in Table

2, are all statistically signi�cant. This result is supported by the larger negative values of the

AIC and SIC statistics reported for the GST models, when compared with the corresponding

SGST models. A comparison of the AIC and SIC statistics reported for all models in Tables

1 and 2 show that the GST(1) model is the best performer. For example, the SIC statistic

falls from -790.964 for SGST(1) to -836.353 for GST(1).

4.2.3 Log-Mixture

The results obtained by using the lognormal mixture option price in (14), are presented in

Table 3. Three versions of the model based on various restrictions on the parameters in (15),

are estimated, referred to respectively as LMIX(1) to LMIX(3). Inspection of Table 3 shows

that all estimated parameters are statistically signi�cant. The signi�cance of the estimate

of �1;4 in (15) for model speci�cation LMIX(1) con�icts with the corresponding results for

this parameter presented in Tables 1 and 2, where the levels e¤ect in the volatility equation

is found to be statistically insigni�cant. The weighting parameter � in (14) is numerically

close to zero meaning that relatively greater weight is given to the second distribution where

volatility is based simply on future currency returns. Nonetheless, the standard errors for

each of the three models show that � is still statistically di¤erent from zero. Furthermore,

consideration of the AIC and SIC statistics leads to the conclusion that the model with the

most general volatility speci�cation, LMIX(1), is the best performer, suggesting that the

maturity variables and the levels e¤ect are indeed important explanatory variables in the

volatility speci�cation when the risk-neutral distribution is based on a mixture of lognormals.

A further comparison of the AIC and SIC statistics across the di¤erent models in Tables 2

and 3 shows that the lognormal mixture models are still inferior to the best performing GST

model.

16

4.2.4 SNP

The semi-nonparametric results using (17) as the option pricing model are presented in Ta-

ble 4. As with the previous models, three volatility speci�cations are entertained, with the

results presented in models SNP(1) to SNP(3). Again, the volatility parameter estimates

highlight the signi�cance of future returns over the life of the option as an important deter-

minant of volatility. The results concerning the importance of the time-to-maturity variables

in determining volatility are mixed as these variables are statistically insigni�cant for the

SNP(1) model, but are just signi�cant for the SNP(2) model. The levels e¤ect in model

SNP(1) is found to be statistically signi�cant, which is in contrast with the results presented

in Tables 1 and 2 for the normal and GST models respectively, but similar to the lognormal

mixture results in Table 3.

The parameter estimates and standard errors of �1 and �2, reported in Table 4, reveal

that nonnormalities are priced in currency options. Comparing the AIC and SIC statistics

across all estimated models in Tables 1 to 4, the semi-nonparametric model is seen to be

inferior to the lognormal mixture model, which, in turn, is inferior to the GST model. The

worst �tting model of all those considered is the GK model.

4.2.5 Summary

The empirical results presented highlight two important properties. First, volatility is time-

varying with signi�cant contributions from the future returns over the life of the option and

the maturity e¤ects. Second, skewness and fatness in the tails of the distribution of currency

returns are priced in options. In particular, the GST distribution in general performs the

best in modelling these nonnormalities.

Comparisons of the Garman-Kohlhagan risk neutral distribution, that is the lognormal

distribution, with the SGST(1), GST(1), LMIX(1) and SNP(1) risk neutral distributions

at the time the option matures, are given in Figure 2. The distributions are based on

St = 165; r = 0:05; i = 0:07 and � = 0:5: The SGST distribution gives less weight to

lower values St+n than the GK distribution, but more weight to values of St+n around the

centre of the distribution at approximately St+n = 165: For values of St+n in the upper tails

of the distributions, the probabilities of the GK and SGST distributions are very similar.

That is, the GK risk neutral distribution is predicting future falls in the exchange rate with

higher probabilities than the SGST distribution, whilst their predictions of future increases

17

in the exchange rate are comparable. The GST gives qualitatively similar results to the

SGST distribution with the positive skewness in this distribution (b� = 0:370 in Table 2)

accentuating the di¤erences with the GK distribution: thinner left tail and more peakedness

in the centre of the distribution. In contrast, the LMIX risk neutral distribution attaches

hardly any weight to values of St+n in the upper tails, with the peak of this distribution

occurring to the right of the GK risk neutral distribution. Finally, the SNP risk neutral

distribution attaches less weight to both tails and greater weight to values of St+n around

St = 165:

The behaviour of the risk neutral distribution over the life of the option is demonstrated in

Figure 3 for the case of the GST(1) distribution. As before the distributions are based on St =

165; r = 0:05; i = 0:07. The range of maturities are � = 0:1; 0:2; � � � 0:5: These distributionsshow that as the option contract approaches maturity, � = 0; the risk neutral distribution

becomes more peaked. This property stems from the estimated volatility function which

according to the parameter estimates reported in Table 2, decreases over time.16

4.3 Forecasting

The model comparisons presented above are all based on within-sample statistical properties.

In this section, following Bakshi, Cao and Chen (1997), Sarwar and Krehbiel (2000) and

Lehar, Scheicher and Schittenkopf (2002), the relative out-of-sample forecasting performance

of the alternative models is investigated. Based on the parameter estimates reported above,

predicted option prices are produced for each contract in the �ve out-of-sample days. Poor

forecasting properties will re�ect evidence of misspeci�cation and/or that the parameter

estimates are not stable over the forecast period.

The forecast errors are assessed using two statistics. The �rst statistic reported is the

16An estimate of the volatility in (3) can be computed at each point in time. To perform this calculation itis necessary to choose a value for the exchange rate at the time of maturity, St+n: The results (not reported)show that volatility decreases from about 9% at the start of the sample period to about 7% at the end of thesample. As a check on the latter volatility estimate a GARCH(1,1) model is estimated using daily currencyreturns over the sample period. The estimated model is

100 (lnSt � lnSt�1) = 0:0241 + etb�2t = 0:0183 + 0:0409e2t�1 + 0:8745b�2t�1:This yields a long-run value of the squared volatility of 0:0183= (1� 0:0409� 0:8745) = 0:2163: The long-run annualised volatility estimate is then

p(250) (0:2163) = 7:354%, which is consistent with the implied

volatility estimate mentioned above.

18

Figure 2: Comparison of the Garman-Kohlhagan risk neutral distribution with the SGST,GST. LMIX and SNP risk neutral distributions.

19

Figure 3: Plot of the GST(1) risk neutral probability distribution for � = 0:1; 0:2; :::; 0:5;maturities. 20

Table 1:

Maximum likelihood estimates of normal option price models for alternative speci�cationsof (3), with standard errors based on the inverse of the hessian in brackets. A value of zero

without a standard error implies that the parameter is set equal to that value.

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

�0 -3.149 -3.010 -2.454 -2.387(0.284) (0.015) (0.003) (0.003)

�1 0.375 0.374 0.430 0.000(0.035) (0.034) (0.001)

�2 1.546 1.550 0.000 0.000(0.048) (0.048)

�3 -0.936 -0.939 0.000 0.000(0.036) (0.036)

�4 0.028 0.000 0.000 0.000(0.056)

lnL=N 0.588 0.588 -0.177 -0.356s2 0.018 0.018 0.083 0.119

AIC(a) -818.871 -820.728 253.771 504.341SIC(b) -796.080 -802.495 262.887 508.900

(a) AIC = -2lnL+2k, where L is the likelihood and k is the number of estimated parameters.

(b) SIC = -2lnL+ln(N)k, where L is the likelihood, N is the sample size and k is the number of estimatedparameters.

21

Table 2:

Maximum likelihood estimates of symmetric GST option price models, SGST(1) toSGST(3), and GST option price models, GST(1) to GST(3), for alternative speci�cationsof (3), with standard errors based on the inverse of the hessian in brackets. A value of zero

without a standard error implies that the parameter is set equal to that value.

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

�0 -3.158 -3.014 -2.570 -5.060 -2.896 -2.601(1.104) (0.015) (0.004) (2.348) (0.022) (0.004)

�1 0.439 0.439 0.586 0.582 0.577 0.757(0.059) (0.056) (0.001) (0.005) (0.002) (0.001)

�2 1.541 1.545 0.000 0.886 1.069 0.000(0.054) (0.049) (0.239) (0.076)

�3 -0.935 -0.938 0.000 -0.452 -0.602 0.000(0.040) (0.036) (0.192 ) (0.056)

�4 0.028 0.000 0.000 0.432 0.000 0.000(0.218) (0.470)

0.561 0.562 0.706 0.625 0.618 0.699(0.078) (0.078) (0.006) (0.062) (0.061) (0.139)

� 0.000 0.000 0.000 0.370 0.438 0.748(0.127) (0.097) (0.171)

lnL=N 0.589 0.589 0.336 0.626 0.615 0.421s2 0.018 0.018 0.030 0.017 0.017 0.025

AIC(a) -818.313 -820.161 -467.104 -868.261 -854.638 -585.268SIC(b) -790.964 -797.370 -453.429 -836.353 -827.289 -567.035

(a) AIC = -2lnL+2k, where L is the likelihood and k is the number of estimated parameters.

(b) SIC = -2lnL+ln(N)k, where L is the likelihood, N is the sample size and k is the number of estimatedparameters.

22

Table 3:

Maximum likelihood estimates of the mixture of lognormal option price model, foralternative speci�cations of (3), with standard errors based on the inverse of the hessian inbrackets. A value of zero without a standard error implies that the parameter is set equal

to that value.

Parameter LMIX(1) LMIX(2) LMIX(3)

�1;0 -0.210 -2.606 -2.589(0.588) (0.047) (0.056)

�1;1 0.527 0.525 0.522(0.013) (0.007) (0.008)

�1;2 -0.560 -0.001 0.000(0.107) (0.020)

�1;3 0.306 0.010 0.000(0.069) (0.019)

�1;4 -0.431 0.000 0.000(0.134)

�2;0 -2.443 -2.600 -2.601(0.062) (0.005) (0.006)

�2;1 -4.108 0.442 0.441(2.079) (0.005) (0.004)

� 0.078 0.049 0.051(0.010) (0.003) (0.002)

lnL=N 0.567 0.546 0.546s2 0.019 0.020 0.020

AIC(a) -818.786 -756.015 -759.959SIC(b) -781.976 -724.108 -737.168

(a) AIC = -2lnL+2k, where L is the likelihood and k is the number of estimated parameters.

(b) SIC = -2lnL+ln(N)k, where L is the likelihood, N is the sample size and k is the number of estimatedparameters.

23

Table 4:

Maximum likelihood estimates of the semi-nonparametric option price models, foralternative speci�cations of (3), with standard errors based on the inverse of the hessian inbrackets. A value of zero without a standard error implies that the parameter is set equal

to that value.

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

�0 -1.015 -2.652 -2.623(0.625) (0.020) (0.006)

�1 0.525 0.522 0.523(0.003) (0.004) (0.001)

�2 -0.056 0.202 0.000(0.070) (0.114)

�3 0.041 -0.196 0.000(0.070) ( 0.090)

�4 -0.313 0.000 0.000(0.122)

�1 -0.244 -0.230 -0.246(0.010) (0.008) (0.004)

�2 0.214 0.193 0.215(0.013) (0.010) (0.006)

lnL=N 0.564 0.556 0.551s2 0.019 0.019 0.019

AIC(a) -780.620 -772.094 -769.141SIC(b) -748.713 -744.745 -750.908

(a) AIC = -2lnL+2k, where L is the likelihood and k is the number of estimated parameters.

(b) SIC = -2lnL+ln(N)k, where L is the likelihood, N is the sample size and k is the number of estimatedparameters.

24

Root Mean Squared Error (RMSE)

RMSE =

s1

N

Xj

fej;t, (22)

where fej;t denotes the forecast error for the jth contract at time t based on a particular

model. The RMSE of the various models are reported in Table 5. The forecast horizon

consists of �ve days beginning June 10th (Wednesday) and ending June 16th (Tuesday),

1998. The RMSE is also reported for all �ve out-of-sample days taken together in the last

column.

The results of the forecasting tests reported in Table 5 show that the GK model yields the

worst forecasts, as it produces the largest RMSE on all �ve days. Allowing for time variation

in volatility based on the currency return until maturity (NORM(3)) yields improvements

in forecasting relative to the GK model. Further gains are obtained by also including time-

to-maturity e¤ects (NORM(2)), but not levels e¤ects (NORM(1)). Allowing for fatness in

the tails of the distribution, but no skewness (SGST), results in even more improvements in

forecastability. Extending the returns distribution to allow for skewness yields improvements

for the simple time-varying volatility model (GST(3)) on all �ve days, and the expanded

volatility model (GST(1) and GST(2)) on days three to �ve, but not on the �rst two days.

The lognormal mixture model (LMIX) produces RMSE�s that are smaller than those of

the GK model, but larger than those associated with the best forecasting models in the

NORM, SGST and GST classes. Interestingly, the three alternative volatility speci�cations

used in the lognormal mixture models produce very similar RMSEs, suggesting that future

currency returns is the main explanatory variable in the volatility equation. Finally, the

semi-nonparametric model (SNP), as with the lognormal mixture model, yields RMSE�s

that tend to be invariant to the volatility speci�cation. In fact, the best forecasting SNP

model is SNP(3), the SNP model with the simplest time-varying volatility speci�cation.

However, this model still does not perform as well as the best forecasting models within the

NORM, SGST and GST speci�cations.

The second statistic used to gauge the relative forecasting performance of the alternative

models is the Diebold and Mariano (1995) (DM) statistic. This statistic is used to assess

the signi�cance of the di¤erence between the forecast errors of any one of the nonnormal

models and the normal model with the same volatility speci�cation. The same statistic is

also calculated to assess the signi�cance of the di¤erence between the normal model with

25

time-varying volatility and the GK model which imposes constant volatility. Denoting by

dj;t the di¤erence between the forecast errors for the jth contract on day t of any particular

model and the forecast errors from the relevant base model, the DM statistic is de�ned as

DM =1N

Pj dj;tq

1N

Pj d

2j;t

: (23)

Under the null hypothesis of no di¤erence between the forecast errors, DM is asymptotically

distributed as N(0; 1):

The results of the Diebold-Mariano test are reported in Table 6. The row corresponding to

the GK model shows that the forecast errors of this model are statistically di¤erent from the

forecast errors obtained from the NORM(1) model. A comparison between the NORM and

symmetric GST models (SGST) reveals a signi�cant di¤erence in the forecasting properties

of the two models for the simplest time-varying volatility speci�cation (SGST(3)), but not for

the more general volatility speci�cations (SGST(1) and SGST(2)). However, the allowance

for skewness in the currency returns distribution (GST) results in signi�cant di¤erences in

forecasting power for all volatility speci�cations. Similar results occur for both lognormal

mixture (LMIX) and semi-nonparametric (SNP) models in most cases.

4.4 Implications for Hedging

4.4.1 Delta Hedging

The empirical results highlight the importance of pricing the nonnormalities in the currency

returns distribution and the time variations in the volatility process. Now the implications

of these departures from the GK model are investigated for constructing hedged portfolios.

In the case of a delta-hedged portfolio, the expression of the delta is given as the derivative of

the price of the call option F , with respect to the exchange rate S:17 Under the assumptions

of constant volatility and normal returns, an analytical expression for the delta for the GK

model is given by (Hull, 2000)

�GK =dF

dS= e�i�N(d1), (24)

where N(�) is the cumulative normal distribution function and d1 is de�ned in (11). In thecase of the option price models based on nonnormal distributions, the delta is computed

17The analysis focuses on delta hedging strategies although it could be extended to constructing vegahedges to hedge against time-variations in volatility.

26

Table 5:

Forecasting performance of competing models across various days(a) based on RMSE using(22), beginning June 10th (Wednesday) and ending June 16th (Tuesday), 1998.

Model Day 1 Day 2 Day 3 Day 4 Day 5 All 5 Days

(4) (5) (4) (8) (10) (31)

GK 0.619 0.597 0.596 0.594 0.559 0.587

NORM(1) 0.035 0.039 0.039 0.058 0.040 0.044NORM(2) 0.035 0.038 0.039 0.058 0.040 0.044NORM(3) 0.505 0.483 0.481 0.476 0.452 0.474

SGST(1) 0.032 0.037 0.036 0.057 0.038 0.043SGST(2) 0.032 0.037 0.036 0.057 0.038 0.043SGST(3) 0.274 0.248 0.243 0.233 0.231 0.242

GST(1) 0.064 0.042 0.025 0.020 0.036 0.038GST(2) 0.066 0.044 0.025 0.021 0.028 0.036GST(3) 0.265 0.241 0.230 0.219 0.218 0.230

LMIX(1) 0.167 0.144 0.139 0.132 0.138 0.142LMIX(2) 0.158 0.135 0.129 0.120 0.131 0.133LMIX(3) 0.157 0.134 0.129 0.119 0.131 0.132

SNP(1) 0.092 0.096 0.070 0.078 0.082 0.083SNP(2) 0.093 0.094 0.072 0.078 0.084 0.084SNP(3) 0.081 0.086 0.064 0.073 0.079 0.077

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

27

Table 6:

Forecasting performance of competing models across various days based on theDiebold-Mariano test using (23), beginning June 10th (Wednesday) and ending June 16th

(Tuesday), 1998.

Model Day 1 Day 2 Day 3 Day 4 Day 5 All 5 Days

(4) (5) (4) (8) (10) (31)

GK -1.978 * -2.212 * -1.982 * -2.811 * -2.934 * -5.408 *

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

SGST(1) -0.659 -0.597 -0.511 -0.454 0.513 -0.540SGST(2) -0.657 -0.597 -0.507 -0.448 0.530 -0.522SGST(3) 1.981 * 2.216 * 1.985 * 2.813 * 2.975 * 5.434 *

GST(1) -1.928 * -2.155 * -1.949 * -2.784 * -2.675 * -5.121 *GST(2) -1.912 * -2.140 * -1.935 * -2.772 * -2.582 * -5.111 *GST(3) 1.990 * 2.225 * 1.993 * 2.822 * 2.914 * 5.415 *

LMIX(1) -1.900 * -2.138 * -1.919 * -2.751 * -2.820 * -5.224 *LMIX(2) -1.981 * -2.214 * -1.983 * -2.811 * -2.939 * -5.403 *LMIX(3) 1.982 * 2.217 * 1.985 * 2.814 * 2.954 * 5.425 *

SNP(1) -1.634 -1.783 * -1.636 -2.344 * -2.561 * -4.502 *SNP(2) -1.569 -1.744 * -1.621 -2.356 * -2.553 * -4.469 *SNP(3) 1.992 * 2.219 * 1.994 * 2.818 * 2.920 * 5.416 *

(a) n.a. = not applicable.

(b) � indicates signi�cance at the 5% level.

28

using a numerical forward di¤erence scheme with a step length of h = 0:001; that is, the

option is evaluated at S and S + h, with the di¤erence divided by h.

The e¤ects of di¤erent distributional assumptions on the delta hedge ratio are highlighted

in Table 7. The calculations are based on a spot exchange rate equal to S = 165, with the

domestic and foreign interest rates set at 5% and 7% respectively. The contract is for a three

month option, � = 0:25, with strike prices ranging from X = 155 to X = 175 in steps of 5,

to allow for a broad moneyness spectrum. These values are based on the data set used in

the empirical illustration.

A comparison of the deltas across the alternative models reveals that the GK model

has uniformly 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, with

the exception of the GST models. This suggests two things. Firstly, modelling skewness

in the returns distribution has important implications for the relationship between the call

option price and the exchange rate. For the present application the function relating the

option price (F ) to the spot price (S) is relatively steeper for the GST models than any

of the other models considered, with delta values around 0:65 compared to values around

0:45. Second, whilst the LMIX and SNP models also price skewness and fatness in the tails

of the returns distribution, given the results concerning within-sample and out-of-sample

performances of the various models above, it is clear that the GST model not only does a

better job in capturing skewness, but that there are important �nancial di¤erences between

the properties of the various nonnormal models. These results suggest that by not pricing

skewness accurately, if at all, the number of foreign currency contracts bought to hedge the

portfolio will be de�cient and that the portfolio is exposed as a consequence.

For the in-the-money options, X = 170 and 175, the results show that the di¤erent

volatility speci�cations now yield di¤erent values for the delta hedge parameter, with the

SGST model yielding the lowest delta hedges and the GST model the highest delta hedges.

Interestingly, the GST delta values for the deep in-the-money option (X = 175) are closer to

the GK hedge value of 0:084 than they are to the in-the-money deltas of either of the other

models.

4.4.2 Risk Management

The implications of constructing delta hedges in the presence of nonnormalities in currency

returns are explored further by performing the following dynamic hedging experiment; see

29

Table 7:

Deltas of competing models across various strike prices: S = 165; r = 0:05; i = 0:07;� = 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) 0.943 0.769 0.442 0.159 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) 1.079 0.947 0.448 0.071 0.001SGST(2) 1.079 0.947 0.447 0.071 0.001SGST(3) 1.086 0.920 0.457 0.099 0.005

GST(1) 1.107 0.961 0.624 0.276 0.076GST(2) 1.128 0.987 0.647 0.288 0.079GST(3) 1.040 0.935 0.666 0.335 0.097

LMIX(1) 0.836 0.691 0.451 0.188 0.035LMIX(2) 0.930 0.752 0.446 0.179 0.048LMIX(3) 0.931 0.752 0.446 0.179 0.048

SNP(1) 0.922 0.774 0.465 0.158 0.023SNP(2) 0.926 0.778 0.472 0.169 0.031SNP(3) 0.927 0.781 0.473 0.164 0.028

30

also Bakshi, Cao and Chen (1997). Consider selling European currency call options of 1000

units of foreign exchange at the strike price X = 165, maturing in 90 days, � = 0:25.18 That

is, the holder of the contract has the right to purchase 1000 units of foreign currency at a

price of 1000 � 165 in domestic currency. Let the domestic and foreign interest rates ber = 5% and i = 7% respectively, and the implied volatility be equal to 9%. The distribution

of z in (12) is assumed to have a GST distribution with = 1 and � = 1: Thus, (2) is used

to simulate the exchange rate over the life of the contract with the time interval equal to

one day. The initial spot rate is S = 165, in which case the option is at-the-money.

Consider setting up a portfolio to minimize the exposure to the call option contract by

using a delta hedge to determine the number of units of foreign assets to buy at the prevailing

exchange rate. The cost of the initial investment in the domestic currency is

I0 = QCF +QSS,

where QC = �1000 represents the number of call options, F is the price of a call option,

QS = �1000 is the total number of units of foreign exchange purchased based on the delta

hedge parameter �, and S is the spot exchange rate. The term QCF is the total price of the

option contract and QSS 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 in either foreign assets being sold or purchased depending on the number of assets that

are needed in the portfolio as determined by the delta hedge. In the case where assets are sold

(bought) the money is invested (borrowed) at the risk free rate of interest r: In computing

the delta hedges, three types are considered depending on the underlying distribution of

currency returns assumed: a normal distribution, a symmetric GST distribution with = 1,

and a skewed GST distribution with = 1 and � = 1:

The results of the dynamic hedging experiment are given in Table 8. For each experiment

the table provides the initial investment of each contract (I0) ; the value of the investment if

I0 is invested at the risk free rate of interest (r = 5%) over the life of the option

I = er90=365I0;

and the value of the portfolio at the time the option matures (V ). A measure of the per-

formance of the portfolio is given by the percentage error between the value of the portfolio18In the case of call options for the British pound, the sizes of contracts in practice are 31,250 pounds

(Hull, 2000).

31

and investing in bonds

Error = 100V � II

:

For a perfectly hedged portfolio the return on the portfolio equals r; so V = I, and hence

Error = 0: An alternative measure of the performance of the portfolio given in Table 8 is

the annual percentage return of the portfolio

rportfolio = 100 ln

�P

I

��365

90

�:

Again, for a perfectly hedged portfolio, rportfolio will equal the risk free rate of interest r: The

experiment is repeated 1000 times with the mean values of each of the quantities reported

in Table 8.

The size of the initital investment to set up the portfolio is smaller for both the deep in-

the-money option (X = 155) and the deep out�of-the-money option (X = 175) when hedging

is based on the GST model. For all hedging models, the percentage hedging error (Error)

is negative showing that the hedged portfolio incurs a loss relative to investing in bonds at

the risk free rate. The size of the hedging error is smallest for the GST model in three of

the �ve contracts but it does not perform as well for the case when X = 160 and for the

deep in-the-money option X = 175: Interestingly, for this option contract, the symmetric

GST (SGST) model has a lower error than the GK model. A similar pattern occurs for the

return on the portfolio. For the at-the-money option (X = 165) ; the loss on the GK hedge

is �2:36 which is larger than the loss incurred on the GST hedged portfolio, �1:87:

5 Conclusions

This paper has provided an alternative framework for pricing currency options which relaxed

the twin assumptions of constant volatility and normal currency returns that underlie the

commonly used pricing model of Garman and Kohlhagen (1983). Time variation in volatility

was achieved by allowing volatility to be a function of the future return over the life of the

option, time-to-maturity e¤ects as well as levels e¤ects. The use of future returns contrasted

with exiting option price models that accommodate time-varying volatility using standard

stochastic volatility or GARCH speci�cations. Nonnormalities in the risk neutral probability

distribution were explicitly modelled using a range of nonnormal distributions that captured

both skewness and fat tails. An important feature of the proposed modelling framework was

that currency options could be priced in a computationally e¢ cient manner, as the pricing

32

Table 8:

Dynamic hedging experiments across various strike prices: S = 165; r = 0:05; i = 0:07;� = 0:25: Currency returns are assumed to be drawn from a GST with parameter = 1

and � = 1: Rebalancing is undertaken daily. Reported values based on mean estimates from1000 Monte Carlo simulations. The initial value of the investment is given by I0; I is thevalue of a portfolio at the time of maturity from investing in bonds and V is the value of

the hedged portfolio at the time of maturity.

Strike Hedging I0 I V Error Rportfolio Rbonds(X) Model (%) (%p:a:) (%p:a:)

155 GK 1374.46 1391.51 1363.86 -1.99 -3.49 5.00SGST 1362.91 1379.82 1352.44 -1.98 -3.54 5.00GST 1359.61 1376.47 1349.89 -1.93 -3.34 5.00

160 GK 1124.23 1138.18 1118.48 -1.73 -2.28 5.00SGST 1109.38 1123.14 1103.47 -1.75 -2.39 5.00GST 1129.18 1143.19 1123.32 -1.74 -2.34 5.00

165 GK 728.90 737.94 724.91 -1.77 -2.36 5.00SGST 733.85 742.95 730.08 -1.73 -2.24 5.00GST 763.55 773.02 760.33 -1.64 -1.87 5.00

170 GK 355.44 359.85 353.51 -1.76 -2.37 5.00SGST 373.59 378.23 371.77 -1.71 -2.13 5.00GST 380.19 384.91 378.37 -1.70 -2.10 5.00

175 GK 127.77 129.35 127.35 -1.54 -1.49 5.00SGST 134.37 136.03 133.99 -1.50 -1.33 5.00GST 112.92 114.32 112.37 -1.70 -2.31 5.00

33

of options entailed only the numerical evaluation of a one-dimensional real integral. This

contrasted with models that require either the use of Monte Carlo simulation to evaluate

option prices, or the use of complex numerical integration methods.

The proposed modelling framework was applied to pricing European currency call options

on the 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 contracts within days as well as across days. The key empirical results showed

that the proposed option price model resulted in large reductions in pricing errors and

improvements in forecasting, compared to the Garman and Kohlhagen (1983) model. Of

the set of nonnormal probability distributions used to model the underlying risk neutral

probability distribution, the generalised Student t distribution performed the best in terms

of capturing skewness and fatness in the tails of the underlying currency returns distribution.

The Garman and Kohlhagen currency option price model performed the worst in terms

of all of the criteria considered. The empirical analysis showed that pricing skewness has

important implications for constructing international hedged portfolios to minimize exposure

to movements in exchange rates. For the present application these results not only showed

that skewness was an important factor in designing the portfolio, but that modelling skewness

via the GST distribution provided a more e¤ective way to do so than using the alternative

nonnormal speci�cations considered in the empirical application. Finally, the results of the

hedging experiment showed that improvements in the hedging capabilities of portfolios could

be enhanced by constructing hedges based on nonnormal distributions.

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