Variance-gamma distribution

name =variance-gamma distribution| type =density| pdf_image =| cdf_image =| parameters =\mu location (real) \alpha (real) \beta asymmetry parameter (real) \lambda 0 shape parameter (alternative parameterizations use \nu=1/\lambda) \gamma = \sqrt{\alpha^2 - \beta^2} 0 | support =x \in (-\infty; +\infty)!| pdf =\frac{\gamma^{2\lambda} | x - \mu|^{\lambda-1/2} K_{\lambda-1/2} \left(\alpha|x - \mu|\right)}{\sqrt{\pi} \Gamma (\lambda)(2 \alpha)^{\lambda-1/2}} ; e^{\beta (x - \mu)}

K_\lambda denotes a modified Bessel function of the second kind \Gamma denotes the Gamma function| cdf =| mean =\mu + 2 \beta \lambda/ \gamma^2| median =| mode =| variance =2\lambda(1 + 2 \beta^2/\gamma^2)/\gamma^2| skewness =|| kurtosis =|| entropy =| mgf =e^{\mu z} \left(\gamma/\sqrt{\alpha^2 -(\beta+z)^2}\right)^{2\lambda}| char =| The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution The variance-gamma distributions form a subclass of the generalised hyperbolic distributions.

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of variance-gamma distributions is closed under convolution in the following sense. If X_1 and X_2 are independent random variables that are variance-gamma distributed with the same values of the parameters \alpha and \beta, but possibly different values of the other parameters, \lambda_1, \mu_1 and \lambda_2, \mu_2, respectively, then X_1 + X_2 is variance-gamma distributed with parameters \alpha, \beta, \lambda_1+\lambda_2 and \mu_1 + \mu_2.

The variance-gamma distribution can also be expressed in terms of three inputs parameters (C,G,M) denoted after the initials of its founders. If the "C", \lambda here, parameter is integer then the distribution has a closed form 2-EPT distribution. See 2-EPT probability density function. Under this restriction closed form option prices can be derived.

If \alpha=1, \lambda=1 and \beta=0, the distribution becomes a Laplace distribution with scale parameter b=1. As long as \lambda=1, alternative choices of \alpha and \beta will produce distributions related to the Laplace distribution, with skewness, scale and location depending on the other parameters.

For a symmetric variance-gamma distribution, the kurtosis can be given by 3(1 + 1/\lambda).

See also Variance gamma process.