2-EPT probability density function

2-EPT Density Function
Parameters	$({\textbf {A}}_{N},{\textbf {b}}_{N},{\textbf {c}}_{N},{\textbf {A}}_{P},{\textbf {b}}_{P},{\textbf {c}}_{P})$ $\mathbb {R} (\sigma ({\textbf {A}}_{P}))<0$ $\mathbb {R} (\sigma ({\textbf {A}}_{N}))>0$
Support	$x\in (-\infty ;+\infty )\!$
PDF	$f(x)=\left\{{\begin{matrix}{\textbf {c}}_{N}e^{{\textbf {A}}_{N}x}{\textbf {b}}_{N}&{\mbox{if }}x<0\\[8pt]{\textbf {c}}_{P}e^{{\textbf {A}}_{P}x}{\textbf {b}}_{P}&{\mbox{if }}x\geq 0\end{matrix}}\right.$
CDF	$F(x)=\left\{{\begin{matrix}{\textbf {c}}_{N}{\textbf {A}}_{N}^{-1}e^{{\textbf {A}}_{N}x}{\textbf {b}}_{N}&{\mbox{if }}x<0\\[8pt]1+{\textbf {c}}_{P}{\textbf {A}}_{P}^{-1}e^{{\textbf {A}}_{P}x}{\textbf {b}}_{P}&{\mbox{if }}x\geq 0\end{matrix}}\right.$
Mean	$-{\textbf {c}}_{N}(-{\textbf {A}}_{N})^{-2}{\textbf {b}}_{N}+{\textbf {c}}_{P}(-{\textbf {A}}_{P})^{-2}{\textbf {b}}_{P}$
CF	$-{\textbf {c}}_{N}(Iiu-{\textbf {A}}_{N})^{-1}{\textbf {b}}_{N}+{\textbf {c}}_{P}(Iiu-{\textbf {A}}_{P})^{-1}{\textbf {b}}_{P}$

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 class of probability density functions on $\mathbb {R}$ with strictly proper rational characteristic functions are referred to as 2-EPT probability density functions. On $[0,\infty )$ as well as $(-\infty ,0)$ these probability density functions are Exponential-Polynomial-Trigonometric (EPT) functions. An EPT density function on $(-\infty ,0)$ can be represented as $f(x)={\textbf {c}}_{N}e^{{\textbf {A}}_{N}x}{\textbf {b}}_{N}$. Similarly the EPT density function on $[0,-\infty )$ is expressed as $f(x)={\textbf {c}}_{P}e^{{\textbf {A}}_{P}x}{\textbf {b}}_{P}$. We have that $({\textbf {A}}_{N},{\textbf {A}}_{P})$ are square matrices, $({\textbf {b}}_{N},{\textbf {b}}_{P})$ column vectors and $({\textbf {c}}_{N},{\textbf {c}}_{P})$ row vectors. $({\textbf {A}}_{N},{\textbf {b}}_{N},{\textbf {c}}_{N},{\textbf {A}}_{P},{\textbf {b}}_{P},{\textbf {c}}_{P})$ is the minimal realization of the 2-EPT function. The general class of probability measures on $\mathbb {R}$ with (proper) rational characteristic functions are densities corresponding to mixtures of the pointmass at zero (``delta distribution") and 2-EPT densities. Unlike phase-type and matrix analytic distributions the 2-EPT probability density functions are defined on the whole real line. It is shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations are illustrated for the two-sided framework in Sexton and Hanzon. The most involved operation is the convolution of 2-EPT densities using state space techniques. Much of the work centers on the ability to decompose the rational characteristic function into the sum of two rational functions with poles located in the open left and resp. open right half plane. The Variance Gamma density is shown to be a 2-EPT density under a parameter restriction and the Variance Gamma asset price process can be implemented to demonstrate the benefits of adopting such an approach for financial modelling purposes. Examples of applications provided include option pricing, computing the Greeks and risk management calculations.

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 $\mu _{1}+\mu _{2}.$

See also Variance gamma process.