Variance-gamma distribution

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variance-gamma distribution
Probability density function
Cumulative distribution function
Parameters μ location (real)
α (real)
β asymmetry parameter (real)
λ > 0
\gamma = \sqrt{\alpha^2 - \beta^2}
Support x \in (-\infty; +\infty)\!
Probability density function (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)}

Kλ denotes a modified Bessel function of the third kind
Γ denotes the Gamma function
Cumulative distribution function (cdf)
Mean μ + 2βλ / γ2
Median
Mode
Variance 2λ(1 + 2β2 / γ2) / γ2
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf) e^{\mu z} \left(\gamma/\sqrt{\alpha^2 -(\beta+z)^2}\right)^{2\lambda}
Characteristic function

The variance-gamma distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The distribution was introduced in the financial literature by Madan and Seneta[1]. 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 X1 and X2 are independent random variable that are variance-gamma distributed with the same values of the parameters α and β, but possibly different values of the other parameters, λ1, μ1 and λ2, μ2, respectively, then X1 + X2 is variance-gamma distributed with parameters α, β,λ1 + λ2 and μ1 + μ2.


  1. ^ D.B. Madan and E. Seneta (1990): The variance gamma (V.G.) model for share market returns, Journal of Business, 63, pp. 511 - 524.
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