Generalised hyperbolic distribution

generalised hyperbolic

Parameters	μ location (real) λ (real) α (real) β asymmetry parameter (real) δ scale parameter (real) $\gamma = \sqrt{\alpha^2 - \beta^2}$
Support	$x \in (-\infty; +\infty)\!$
PDF	$\frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)} \; e^{\beta (x - \mu)} \!$ $\times \frac{K_{\lambda - 1/2}\left(\alpha \sqrt{\delta^2 + (x - \mu)^2}\right)}{\left(\sqrt{\delta^2 + (x - \mu)^2} / \alpha\right)^{1/2 - \lambda}} \!$
Mean	$\mu + \frac{\delta \beta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)}$
Variance	$\frac{\delta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)} + \frac{\beta^2\delta^2}{\gamma^2}\left( \frac{K_{\lambda+2}(\delta\gamma)}{K_{\lambda}(\delta\gamma)} - \frac{K_{\lambda+1}^2(\delta\gamma)}{K_{\lambda}^2(\delta\gamma)} \right)$
MGF	$\frac{e^{\mu z}\gamma^\lambda}{(\sqrt{\alpha^2 -(\beta +z)^2})^\lambda} \frac{K_\lambda(\delta \sqrt{\alpha^2 -(\beta +z)^2})}{K_\lambda (\delta \gamma)}$

The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution. Its probability density function (see the box) is given in terms of modified Bessel function of the second kind, denoted by K_λ.

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

It is mainly applied to areas that require sufficient probability of far-field behaviour, which it can model due to its semi-heavy tails—a property the normal distribution does not possess. The generalised hyperbolic distribution is often used in economics, with particular application in the fields of modelling financial markets and risk management, due to its semi-heavy tails. This class is closed under linear operations. It was introduced by Ole Barndorff-Nielsen.

Related distributions

• $X \sim \mathrm{GH}(-\frac{\nu}{2}, 0, 0, \sqrt{\nu}, \mu)\,$ has a Student's t-distribution with ν 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 (NIG).
• $X \sim \mathrm{GH}(?, ?, ?, ?, ?)\,$ normal-inverse chi-squared distribution
• $X \sim \mathrm{GH}(?, ?, ?, ?, ?)\,$ normal-inverse gamma distribution (NI)

• $X \sim \mathrm{GH}(\lambda, \alpha, \beta, 0, \mu)\,$ has a variance-gamma distribution.