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 class of probability density functions on $\mathbb {R}$ with strictly proper rational characteristic function 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 geometric 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 distribution density is shown to be a 2-EPT density under a parameter restriction and the Variance gamma 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. Fitting 2-EPT density functions to empirical data has also been considered. It can be shown using Parsevals Theorem and a isometry that approximating the discrete time rational transform is equivalent to approximating the 2-EPT density itslef in the L-2 Norm sense. The rational approximation software RARL2 is used to approximate the discrete time rational characteristic function of the density .