The distribution of exchange rate returns and the pricing of currency options
Introduction
There now exists a number of stylized facts which have arisen out of empirical studies of nominal exchange rate returns. First, the unconditional distribution of exchange rate returns is leptokurtic, as the distribution tends to exhibit a sharper peak and fatter tails relative to the normal distribution; see for example, Friedman and Vandersteel, 1982, Boothe and Glassman, 1987, Koedijk et al., 1990, and Koedijk et al. (1992). This property is demonstrated in Fig. 1, which gives the empirical distribution of four daily exchange rate returns over the period 1984 to 1991, compared with the normal distribution.1 Second, exchange rate returns exhibit temporal dependencies as a result of volatility clustering. Returns exhibit periods of turbulence, where large changes are followed by large changes, as well as periods of tranquillity, with small changes followed by small changes (Baillie and Bollerslev, 1990). Third, nontrading periods contribute to volatility when markets reopen at the start of a new week, or after a holiday period (Baillie and Bollerslev, 1989). Fourth, de Vries (1994)has discussed the conditions which may result in the distribution of exchange rate returns becoming skewed; see also Engle and González-Rivera (1991).
The occurrence of volatility clustering in exchange rate returns is an important contributing factor to the observed leptokurtosis in the unconditional distribution, as the peakedness in the distribution reflects periods of tranquillity when there is very little movement in the exchange rate, whilst the fatness in the tails identifies periods of turbulence when the exchange rate exhibits large movements. This observation has led to the construction of empirical models of exchange rate returns which allow for conditional heteroscedasticity based on the ARCH framework of Engle (1982)as well as the GARCH and EGARCH generalizations suggested by Bollerslev (1986)and Nelson (1991)respectively; see Bollerslev et al. (1992)and Bollerslev et al. (1994), for a recent review of the ARCH class of models, as well as Gallant et al. (1991)who give a formal motivation for using ARCH conditional variance models.
In many of the earlier empirical studies on exchange rate returns which incorporate an ARCH conditional variance structure, the conditional distribution is assumed to be normal. However, while the conditional variance specification helps to capture some of the observed leptokurtosis in the unconditional distribution of exchange rate returns, it does not explain all of it; see for example Hsieh (1989)and Baillie and Bollerslev (1990). This has led to the development of more general models which replace the normal distribution with various nonnormal conditional distributions.2
The aim of this paper is to formulate and estimate an empirical model of daily exchange rate returns for four bilateral currencies: $Australian/$US ($A/$US), Yen/$US (Yen/$US), Deutchmark/$US (DM/$US), and Sterling/$US (Stg/$US). The model is based on a conditional generalized Student t distribution with ARCH-like conditional variance structures. The generalized Student t is derived in Lye and Martin (1993)and has the advantage that it can exhibit a range of distributional shapes including skewness and kurtosis; see also Lye and Martin (1994). One important feature of the generalized Student t distribution is that it nests a number of nonnormal distributions, thereby facilitating the use of standard hypothesis testing procedures based on Lagrange multiplier test statistics to discriminate between alternative distributional models. The statistical model is shown to be flexible enough to capture the temporal dependencies and hence the observed leptokurtosis in exchange rate returns.
An important implication of the derived model is that it not only highlights potential misspecification problems associated with models based on either normality or even the Student t distribution, but it also points to a solution and hence a direction for future research in exchange rate theory. In particular, acceptance of the hypothesis that the distribution is generalized Student t implies that the underlying dynamics of exchange rate returns is governed by a nonlinear stochastic differential equation.
Implications of the framework are also investigated for pricing currency options with the result that a failure to account adequately for observed skewness and leptokurtosis in the conditional distribution can lead to mispriced options. The degree of mispricing when volatility over the life of the contract is not assumed to be constant, is also investigated.
The rest of the paper proceeds as follows. An empirical model of the conditional distribution of exchange rate returns is formulated in Section 2. Estimation results, diagnostics and distributional specification tests resulting from the application of this model to four currencies are presented in Section 3. Implications of the analysis for the pricing of currency options are examined in Section 4, while Section 5contains the main conclusions of the paper.
Section snippets
An empirical model of the distribution of exchange rate returns
In this section, a model of the distribution of exchange rate returns is derived which is flexible enough to model a range of nonnormal unconditional distribution characteristics. An important property of this class of distributions is that it embeds many of the existing empirical models of the distribution of exchange rate returns.
Let et represent the exchange rate return, which is assumed to have zero mean and time-varying variance ht. The standardized exchange rate return, defined as
Application
The model of exchange rate returns with the conditional distribution given by the generalized Student t distribution in (5), and the conditional variance specifications given in (6), is now estimated using maximum likelihood procedures for the four daily bilateral exchange rates over the period 1984 to 1991 presented in Fig. 1. As the generalized Student t distribution is a member of the generalized exponential family, the maximum likelihood estimator has the property that it is best
Implications for pricing currency options
One important reason for correctly identifying the underlying distribution governing movements in exchange rate returns concerns the pricing of currency options. Currency options priced using Black and Scholes formulae assume that the exchange rate is lognormally distributed with a constant variance. However, a number of studies have found that using this approach tends to misprice options. For example, Bodurtha and Courtadon (1987)find that the Black–Scholes model under-prices out-of-the money
Conclusions
In this paper, an empirical model of the distribution of exchange rate returns has been formulated and applied to the $A/$US, Yen/$US, DM/$US and Stg/$US daily bilateral exchange rates over the period 1984 to 1991. The model combines a conditional variance specification with a conditional distribution based on the generalized Student t distribution. An advantage of this conditional distribution is that it can capture a range of distributional characteristics such as leptokurtosis and skewness.
Acknowledgements
We would like to thank a Co-Editor and an Associate editor as well as two referees for very helpful and insightful comments on a previous version of the paper. Special thanks to Mardi Dungey and the Reserve Bank of Australia for providing the data.
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