2-EPT probability density function

In probability theory, a 2-EPT probability density function is a class of probability density functions on the real line. The class contains the density functions of all distributions that have characteristic functions that are strictly proper rational functions (i.e., the degree of the numerator is strictly less than the degree of the denominator).

2-EPT Density Function

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

Definition

A 2-EPT probability density function is a probability density function on ${\displaystyle \mathbb {R} }$ with a strictly proper rational characteristic function. On either ${\displaystyle [0,+\infty )}$ or ${\displaystyle (-\infty ,0)}$ these probability density functions are exponential-polynomial-trigonometric (EPT) functions.

Any EPT density function on ${\displaystyle (-\infty ,0)}$ can be represented as

${\displaystyle f(x)={\textbf {c}}_{N}e^{{\textbf {A}}_{N}x}{\textbf {b}}_{N},}$

where e represents a matrix exponential, ${\displaystyle ({\textbf {A}}_{N},{\textbf {A}}_{P})}$ are square matrices, ${\displaystyle ({\textbf {b}}_{N},{\textbf {b}}_{P})}$ are column vectors and ${\displaystyle ({\textbf {c}}_{N},{\textbf {c}}_{P})}$ are row vectors. Similarly the EPT density function on ${\displaystyle [0,-\infty )}$ is expressed as

${\displaystyle f(x)={\textbf {c}}_{P}e^{{\textbf {A}}_{P}x}{\textbf {b}}_{P}.}$

The parameterization ${\displaystyle ({\textbf {A}}_{N},{\textbf {b}}_{N},{\textbf {c}}_{N},{\textbf {A}}_{P},{\textbf {b}}_{P},{\textbf {c}}_{P})}$ is the minimal realization[1] of the 2-EPT function.

The general class of probability measures on ${\displaystyle \mathbb {R} }$ 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[2] distributions, the 2-EPT probability density functions are defined on the whole real line. It has been shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations have been illustrated for the two-sided framework in Sexton and Hanzon.[3] 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 either the open left or open right half plane. The variance-gamma distribution density has been shown to be a 2-EPT density under a parameter restriction and the variance gamma process[4] can be implemented to demonstrate the benefits of adopting such an approach for financial modelling purposes.

It can be shown using Parseval's theorem and an isometry that approximating the discrete time rational transform is equivalent to approximating the 2-EPT density itself in the L-2 Norm sense. The rational approximation software RARL2 is used to approximate the discrete time rational characteristic function of the density.[5]

Applications

Examples of applications include option pricing, computing the Greeks and risk management calculations. Fitting 2-EPT density functions to empirical data has also been considered.[6]

Notes

1. Kailath, T. (1980) Linear Systems, Prentice Hall, 1980
2. Neuts, M. "Probability Distributions of Phase Type", Liber Amicorum Prof. Emeritus H. Florin pages 173-206, Department of Mathematics, University of Louvain, Belgium 1975
3. Sexton, C. and Hanzon,B.,"State Space Calculations for two-sided EPT Densities with Financial Modelling Applications", www.2-ept.com
4. Madan, D., Carr, P., Chang, E. (1998) "The Variance Gamma Process and Option Pricing", European Finance Review 2: 79–105
5. Olivi, M. (2010) "Parametrization of Rational Lossless Matrices with Applications to Linear System Theory", HDR Thesis
6. Sexton, C., Olivi, M., Hanzon, B, "Rational Approximation of Transfer Functions for Non-Negative EPT Densities", Draft paper

External links

• 2 - Exponential-Polynomial-Trigonometric (2-EPT) Probability Density Functions Website for background and Matlab implementations