Generalised hyperbolic distribution

The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function


 * $$f(x) = \frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)}\;

\frac{K_{\lambda - 1/2}\left(\alpha \sqrt{\delta^2 + (x - \mu)^2}\right)}{\left(\sqrt{\delta^2 + (x - \mu)^2} / \alpha\right)^{1/2 - \lambda}}\; e^{\beta (x - \mu)}$$

where $$K_\nu$$ is the modified Bessel function of the second kind.

As the name suggests it is of a very general form, being the superclass of, among others, the Student's t-distribution, the hyperbolic distribution and the normal-inverse Gaussian distribution.

Its main areas of application are those which require sufficient probability of far-field behaviour, which it can model due to its semi-heavy tails, a property that the normal distribution does not possess. The generalised hyperbolic distribution is well-used in economics, with particular application in the fields of modelling financial markets and risk management, due to its semi-heavy tails.

Related distributions

 * $$X \sim \mathrm{GH}(-\frac{\nu}{2}, 0, 0, \sqrt{\nu}, \mu)$$ has a Student's t-distribution with $$\nu$$ degrees of freedom.
 * $$X \sim \mathrm{GH}(1, \alpha, \beta, \delta, \mu)$$ has a hyperbolic distribution.
 * $$X \sim \mathrm{GH}(-1/2, \alpha, \beta, \delta, \mu)$$ has a normal-inverse Gaussian distribution.