Variance-gamma distribution

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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.

Notes

↑ 1.0 1.1 Kotz, S. et al (2001). The Laplace Distribution and Generalizations. Birkhauser. 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

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, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
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
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled 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 Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete 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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