Variance-gamma distribution

variance-gamma distribution
Parameters	$\mu$ location (real) $\alpha$ (real) $\beta$ asymmetry parameter (real) $\lambda >0$ $\gamma = \sqrt{\alpha^2 - \beta^2} > 0$
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
Mean	$\mu + 2 \beta \lambda/ \gamma^2$
Variance	$2\lambda(1 + 2 \beta^2/\gamma^2)/\gamma^2$
MGF	$e^{\mu z} \left(\gamma/\sqrt{\alpha^2 -(\beta+z)^2}\right)^{2\lambda}$

The variance-gamma distribution, generalized Laplace distribution^[1] or Bessel function distribution^[1] is a 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.^[2] 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.

Differential equation

The pdf of the variance-gamma distribution is a solution of the following differential equation for $x>u$ :

\left\{\begin{array}{l} (x-\mu ) f''(x)-2 f'(x) (-\beta\mu+\lambda+\beta x-1)+ f(x) \left(\alpha^2 \mu-\beta (\beta\mu-2 \lambda+2)+ x \left(\beta^2-\alpha^2\right)\right)=0, \\[12pt] f(0)=\frac{\sqrt{\alpha} \left(-\frac{1}{2}\right)^{\lambda-\frac{1}{2}} e^{-\beta\mu} \mu^{\lambda-\frac{1}{2}} \left(\alpha-\frac{\beta^2}{\alpha}\right)^{\lambda} K_{\lambda-\frac{1}{2}}(-\alpha\mu)}{\sqrt{\pi} \Gamma(\lambda)}, \\[12pt] f'(0)=\frac{\sqrt{\alpha} 2^{\frac{1}{2}-\lambda} \mu e^{-\beta\mu} (-\mu )^{\lambda- \frac{5}{2}} \left(\alpha-\frac{\beta^2}{\alpha}\right)^{\lambda} \left((\beta\mu-2 \lambda+1) K_{\lambda -\frac{1}{2}}(-\alpha\mu)- \alpha\mu K_{\lambda+\frac{1}{2}}(-\alpha\mu)\right)}{\sqrt{\pi} \Gamma(\lambda)} \end{array}\right\}

It is a solution of the following differential equation for $x<u$ :

\left\{\begin{array}{l} (x-\mu ) f''(x)-2 f'(x) (-\beta\mu+\lambda+\beta x-1)+ f(x) \left(\alpha^2 \mu-\beta(\beta\mu-2 \lambda+2)+ x \left(\beta^2-\alpha^2\right)\right)=0, \\[12pt] f(0)=\frac{2^{\frac{1}{2}-\lambda}\sqrt{\frac{\alpha }{\mu}} e^{-\beta\mu} \left(\mu \left(\alpha- \frac{\beta^2}{\alpha}\right)\right)^{\lambda} K_{\lambda-\frac{1}{2}}(\alpha\mu)}{\sqrt{\pi} \Gamma(\lambda)}, \\[12pt] f'(0)=\frac{\sqrt{\alpha} 2^{\frac{1}{2}-\lambda} e^{-\beta\mu} \mu^{\lambda-\frac{3}{2}} \left(\alpha-\frac{\beta^2}{\alpha}\right)^{\lambda} \left((\beta\mu-2 \lambda+1) K_{\lambda-\frac{1}{2}}(\alpha\mu)+\alpha\mu K_{\lambda+\frac{1}{2}}(\alpha\mu)\right)}{\sqrt{\pi} \Gamma (\lambda)} \end{array}\right\}

Notes

1 2 Kotz, S.; et al. (2001). The Laplace Distribution and Generalizations. Birkhäuser. p. 180. ISBN 0-8176-4166-1.
↑ D.B. Madan and E. Seneta (1990): The variance gamma (V.G.) model for share market returns, Journal of Business, 63, pp. 511–524.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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