View source for Variance-gamma distribution

{{Probability distribution |
  name       =variance-gamma distribution|
  type       =density|
  pdf_image  =|
  cdf_image  =|
  parameters =<math>\mu</math> [[Location parameter|location]] ([[Real number|real]])<br/><math>\alpha</math> <!--to do--> (real)<br/><math>\beta</math> asymmetry parameter (real)<br/><math>\lambda > 0</math> shape parameter (alternative parameterizations use <math>\nu=1/\lambda</math><ref name=nestler/>)<br/><math>\gamma = \sqrt{\alpha^2 - \beta^2} > 0 </math>|
  support    =<math>x \in (-\infty; +\infty)\!</math>|
  pdf        =<math>\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)}</math> <br/><br/><math>K_\lambda</math> denotes a modified Bessel function of the second kind<br/><math>\Gamma</math> denotes the [[Gamma function]]|
  cdf        =<!-- to do -->|
  mean       =<math>\mu + 2 \beta \lambda/ \gamma^2</math>|
  median     =<!-- to do -->|
  mode       =|
  variance   =<math>2\lambda(1 + 2 \beta^2/\gamma^2)/\gamma^2</math>|
  skewness   =<!-- to do -->||
  kurtosis   =<!-- to do -->||
  entropy    =<!-- to do -->|
  mgf        =<math>e^{\mu z} \left(\gamma/\sqrt{\alpha^2 -(\beta+z)^2}\right)^{2\lambda}</math>|
  char       =<!-- to do -->|
}}
The '''variance-gamma distribution''', '''generalized Laplace distribution'''<ref name=laplace>{{cite book|title=The Laplace Distribution and Generalizations|url=https://archive.org/details/laplacedistribut00kotz|url-access=limited|author=Kotz, S.|page=[https://archive.org/details/laplacedistribut00kotz/page/n194 180]|year=2001|publisher=Birkhäuser|isbn=0-8176-4166-1|display-authors=0}}</ref> or '''Bessel function distribution'''<ref name=laplace/> is a continuous probability distribution that is defined as the [[Normal variance-mean mixture|normal variance-mean mixture]] where the mixing density is the [[Gamma distribution|gamma distribution]]. The tails of the distribution decrease more slowly than the [[Normal distribution|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.<ref>D.B. Madan and E. Seneta (1990): The variance gamma (V.G.) model for share market returns, ''Journal of Business'', 63, pp. 511–524.</ref> The variance-gamma distributions form a subclass of the [[Generalised hyperbolic distribution|generalised hyperbolic distribution]]s.

The fact that there is a simple expression for the moment generating function implies that simple expressions for all [[Moment (mathematics)|moments]] are available. The class of variance-gamma distributions is closed under [[Convolution|convolution]] in the following sense. If <math>X_1</math> and <math>X_2</math> are independent [[Random variable|random variable]]s that are variance-gamma distributed with the same values of the parameters <math>\alpha</math> and <math>\beta</math>, but possibly different values of the other parameters,  <math>\lambda_1</math>, <math>\mu_1</math> and <math>\lambda_2,</math> <math>\mu_2</math>, respectively, then  <math>X_1 + X_2</math> is variance-gamma distributed with parameters <math>\alpha</math>, <math>\beta</math>, <math>\lambda_1+\lambda_2</math> and <math>\mu_1  + \mu_2</math>.

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", <math>\lambda</math> here, parameter is integer then the distribution has a closed form 2-EPT distribution. See [[2-EPT probability density function|2-EPT probability density function]]. Under this restriction closed form option prices can be derived.

If <math>\alpha=1</math>, <math>\lambda=1</math> and <math>\beta=0</math>, the distribution becomes a [[Laplace distribution]] with [[Scale parameter|scale parameter]] <math>b=1</math>.  As long as <math>\lambda=1</math>, alternative choices of <math>\alpha</math> and <math>\beta</math> will produce distributions related to the Laplace distribution, with skewness, scale and location depending on the other parameters.<ref>{{cite book|title=Complex Systems in Finance and Econometrics|url=https://archive.org/details/complexsystemsfi00meye|url-access=limited|author=Meyers, Robert A.|page=[https://archive.org/details/complexsystemsfi00meye/page/n349 326]|year=2010|publisher=Springer|isbn=9781441977007}}</ref>

For a symmetric variance-gamma distribution, the [[Kurtosis|kurtosis]] can be given by <math>3(1 + 1/\lambda)</math>.<ref name=nestler>{{cite journal|title=The variance gamma distribution|author1=Nestler, Scott | author2= Hall, Andrew|doi=10.1111/j.1740-9713.2019.01314.x|date=October 4, 2019|journal=The Royal Statistical Society|volume=16 |issue=5 |pages=10–11 |doi-access=free}}</ref>

See also [[Variance gamma process]].

== Notes ==
<references/>

{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Variance-Gamma Distribution}}
[[Category:Continuous distributions]]
[[Category:Infinitely divisible probability distributions]]

{{Sourceattribution|Variance-gamma distribution}}