Tail Behaviour and Tail Dependence of Generalized Hyperbolic Distributions

Generalized hyperbolic distributions have been well established in finance during the last two decades. However, their application, in particular the computation of distribution functions and quantiles, is numerically demanding. Moreover, they are, in general, not stable under convolution which makes the computation of quantiles in factor models driven by these distributions even more complicated. In the first part of the present paper, we take a closer look at the tail behaviour of univariate generalized hyperbolic distributions and their convolutions and provide asymptotic formulas for the quantile functions that allow for an approximate calculation of quantiles for very small resp. large probabilities. Using the latter, we then analyze the dependence structure of multivariate generalized hyperbolic distributions. In particular, we concentrate on the implied copula and determine its tail dependence coefficients. Our main result states that the generalized hyperbolic copula can only attain the two extremal values 0 or 1 for the latter, that is, it is either tail independent or completely dependent. We provide necessary conditions for each case to occur as well as a simpler criterion for tail independence. Possible limit distributions of the generalized hyperbolic family are also included in our investigations.