# Financial Modelling with 2-EPT Levy Processes

The class of probability density functions on R with strictly proper rational characteristic functions are considered. On the positive half-line as well as on the negative half-line these probability density functions are Exponential-Polynomial-Trigonometric (EPT) functions which we abbreviate as 2-EPT densities. EPT density functions can be represented as f(x)=c.exp(Ax).b, where ``A” is a square matrix, ``b” a column vector and ``c” a row vector. The triple (A,b,c) is called a realization of the EPT density function. The more general class of probability measures on R with (proper) rational characteristic functions is also considered whose densities correspond to mixtures of the pointmass at zero (``delta distribution”) and 2-EPT densities. The well-known Variance Gamma density is shown to be a 2-EPT density under a parameter restriction and we implement the Variance Gamma asset price process to demonstrate the benefits of adopting such an approach for financial modelling purposes. Variance Gamma processes are Levy processes. We give conditions under which a 2-EPT distribution is infinitely divisible and gives rise to a Levy process. Here we make use of recent results on sufficient conditions for an EPT function to be non-negative (cf ,). In this way we arrive at a rich class of Levy processes for which there are closed form formulae for many option prices and their corresponding Greeks. Value-at-Risk computations are also straightforward in this framework.

 B. Hanzon, F. Holland, “Non-negativity Analysis for Exponential-Polynomial-Trigonometric Functions on [0,infinity),” to appear in: Proceedings of IWOTA 2010 , Operator Theory: Advances and Applications (OT),Birkhaeuser Verlag, Basel, Boston, Berlin; an earlier version of the paper can be downloaded from www.edgeworth.biz

 B. Hanzon, F. Holland, “Non-negativity of Exponential Polynomial Trigonomet- ric Functions-a Budan Fourier sequence approach”, Poster 429 presented at the Bachelier Finance Society Congress, Toronto, 2010; available on-line at web-address http://euclid.ucc.ie/sta_/hanzon/BFSTorontoPosterHanzonHollandFinal.pdf

Papers and software also available at www.2-ept.com

Key Words: Variance Gamma; Rational Characteristic Functions; Non-Negativity of EPT Functions; Option Pricing; Levy Processes

This talk is part of the CUED Control Group Seminars series.