The Generalised Hyperbolic Skew Student’s t-distribution

The empirical distribution of daily returns from financial market variables such as exchange
rates, equity prices, and interest rates, is often skewed, having one heavy, and one semiheavy,
or more Gaussian-like tail. The NIG distribution, that has two semi-heavy tails, models
skewness rather well, but only in cases where the tails are not too heavy. On the other hand,
the skew Student’s t-distributions presented in the literature have two polynomial tails. Hence,
they fit heavy-tailed data well, but they do not handle substantial skewness.
In this paper, we argue for a special case of the generalised hyperbolic distribution that we
denote the GH skew Student’s t-distribution. This distribution has the important property that
one tail has polynomial, and the other exponential behaviour. Further, it is the only subclass of
the generalised hyperbolic distribution having this property. Although the GH skew Student’s tdistribution
has been previously proposed in the literature, it is not well known, and specifically,
its special tail behaviour has not been addressed.
This paper presents empirical evidence of exponential/polynomial tail behaviour in skew financial
data, and demonstrates the superiority of the GH skew Student’s t-distribution with respect
to data fit, compared with its competitors. Through VaR and expected shortfall calculations we
show why the exponential/polynomial tail behaviour is important in practice. We also present a
simple algorithm for computing the MLE estimators, using a mixture representation of the GH
skew Student’s t-distribution and the EM-algorithm.