Modelling Financial Data Using Generalized Hyperbolic Distributions

This note describes estimation algorithms for generalized hyperbolic hyperbolic and nor mal inverse Gaussian distributions These distributions provide a better t to empirically observed log return distributions of nancial assets than the classical normal distributions Based on the better t to the semi heavy tails of nancial assets we can compute more realistic Value at Risk estimates The modelling of nancial assets as stochastic processes is determined by distributional assump tions on the increments and the dependence structure It is well known that the returns of most nancial assets have semi heavy tails i e the actual kurtosis is higher than the zero kurtosis of the normal distribution see Pagan On the other hand the use of stable distributions leads to models with nonexisting moments The class of generalized hyperbolic distributions and its sub classes the hyperbolic and the normal inverse Gaussian distributions possess these semi heavy tails Generalized hyperbolic distributions were introduced by Barndor Nielsen and applied e g to model grain size distributions of wind blown sands The mathematical properties of these distributions are well known see Barndor Nielsen Bl sild Recently generalized hyperbolic distributions resp their sub classes were proposed as a model for the distribution of increments of nancial price processes see Eberlein Keller Rydberg Barndor Nielsen Eber lein Keller Prause and as limit distributions of di usions see Bibby S rensen Nevertheless studies were only published concerning the estimation and application to nancial data in the special case of hyperbolic distributions In this study we present parameter estima tions for German stock and US stock index data and evaluate the goodness of t In particular we look at the tails of the distributions Generalized Hyperbolic Distributions Generalized hyperbolic GH distributions are given by the Lebesgue density gh x a x K p x exp x a a p K p x R j j if j j if j j if where K is a modi ed Bessel function The parameters and describe the location and the scale of the distribution Note that this distribution may be represented as a normal variance mean mixture with the generalized inverse Gaussian as mixing distribution see Barndor Nielsen Bl sild The normal distribution is obtained as a limiting case for and see Barndor Nielsen Generalized hyperbolic distributions are in nitely divisible hence they generate a L evy processes see Barndor Nielsen Halgreen Eberlein Keller Using the properties of Bessel functions K it is possible to simplify the function gh whenever or For we get the normal inverse Gaussian NIG distribution nig x exp p x K p x p x x R j j and for the hyperbolic distribution HYP