Variance-gamma distribution

The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution 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. The variance-gamma distributions form a subclass of the generalised hyperbolic distributions. The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution 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. 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 {displaystyle X_{1}} and X 2 {displaystyle X_{2}} are independent random variables that are variance-gamma distributed with the same values of the parameters α {displaystyle alpha } and β {displaystyle eta } , but possibly different values of the other parameters, λ 1 {displaystyle lambda _{1}} , μ 1 {displaystyle mu _{1}} and λ 2 , {displaystyle lambda _{2},} μ 2 {displaystyle mu _{2}} , respectively, then X 1 + X 2 {displaystyle X_{1}+X_{2}} is variance-gamma distributed with parameters α {displaystyle alpha } , β {displaystyle eta } , λ 1 + λ 2 {displaystyle lambda _{1}+lambda _{2}} and μ 1 + μ 2 {displaystyle 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', λ {displaystyle 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 α = 1 {displaystyle alpha =1} , λ = 1 {displaystyle lambda =1} and β = 0 {displaystyle eta =0} , the distribution becomes a Laplace distribution with scale parameter b = 1 {displaystyle b=1} . As long as λ = 1 {displaystyle lambda =1} , alternative choices of α {displaystyle alpha } and β {displaystyle eta } will produce distributions related to the Laplace distribution, with skewness, scale and location depending on the other parameters. See also Variance gamma process.