Variance-gamma distribution

Parameters $\mu$ location (real)$\alpha$ (real)$\beta$ asymmetry parameter (real)$\lambda >0$ $\gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}>0$ $x\in (-\infty ;+\infty )\!$ ${\frac {\gamma ^{2\lambda }|x-\mu |^{\lambda -1/2}K_{\lambda -1/2}\left(\alpha |x-\mu |\right)}{{\sqrt {\pi }}\Gamma (\lambda )(2\alpha )^{\lambda -1/2}}}\;e^{\beta (x-\mu )}$ $K_{\lambda }$ denotes a modified Bessel function of the second kind$\Gamma$ denotes the Gamma function $\mu +2\beta \lambda /\gamma ^{2}$ $2\lambda (1+2\beta ^{2}/\gamma ^{2})/\gamma ^{2}$ $e^{\mu z}\left(\gamma /{\sqrt {\alpha ^{2}-(\beta +z)^{2}}}\right)^{2\lambda }$ 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.