Abstract
A generalization of the hyperbolic secant distribution which allows for both skewness and leptokurtosis was given by Morris (1982). Recently, Vaughan (2002) proposed another flexible generalization of the hyperbolic secant distribution which has a lot of nice properties but is not able to allow for skewness. For this reason, Fischer and Vaughan (2002) additionally introduced a skewness parameter by means of splitting the scale parameter and showed that most of the nice properties are preserved. We briefly reviewthis class of distributions and apply them to financial return data. By means of the Nikkei225 data, it will be shown that this class of distributions, the socalled skew generalized secant hyperbolic distribution, provides an excellent fit in the context of unconditional and conditional return models.
Highlights
The hyperbolic secant distribution, which was first studied by Baten (1934) and Talacko (1956), seems to be an appropriate candidate as a starting point for financial return models
As the cumulative distribution function of the generalized secant hyperbolic (GSH) distribution is explicitly known, Fischer and Vaughan (2002) decided in favour of splitting the scale parameter, as it was done by Fernandez et al (1995) for the skewed exponential power distribution
Two generalizations of the hyperbolic secant distribution have been proposed in the last years which seem to be encouraging as model for financial return data: The natural exponential families (NEF)-generalized hyperbolic secant (GHS) distribution of Morris (1982) and the skewed generalized secant hyperbolic distribution (SGSH) distribution of Fischer and Vaughan (2002)
Summary
The hyperbolic secant distribution, which was first studied by Baten (1934) and Talacko (1956), seems to be an appropriate candidate as a starting point for financial return models. The cumulative distribution function admits a closed form implying that, for example, risk neutral probabilities of option prices can be calculated fast and accurate This distribution is reproductive (i.e., the class is preserved under convolution), infinitely divisible with existing moment-generating function and has finite moments. The moment-generating function and all moments exist, and the cumulative distribution is given in closed form This family does not allow for skewness. It will be shown that this family, termed as skewed generalized secant hyperbolic distribution (SGSH), provides an excellent fit to the Nikkei225 data This is verified in the context of unconditional and conditional return models. We compare the results to other popular models for financial return data which have been proposed in the literature in the past: The α-stable distributions (see, e.g., Mittnik et al, 1998), the class of generalized hyperbolic distributions (see, e.g., Prause, 1999), the generalized logistic family of McDonald (1991) and a skewed generalized family of t-distributions of Grottke (2001)
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