Numerical solution of non-local PDEs arising in Finance.

Abstract

It is a well known fact that the value of an option on an asset following a Levy jump-process, can be found by solving a Partial Integro-Differential Equation (PIDE). In this project, two new schemes are presented to solve these kinds of PIDEs when the underlying Levy process is of infinite activity. The infinite activity jump-process leads to a singular Levy measure, which has important numerical ramifications and needs to be handled with care. The schemes presented calculate the non-local integral operator via a fast Fourier transform (FFT), and an explicit/implicit operator splitting scheme of the local/global operators is performed. Both schemes will be of 2nd order on a regular Levy measure, but the singularity degrades convergence to lie in between 1st and 2nd order depending on the singularity strength. On the logarithmically transformed PIDE, the schemes are proven to be consistent, monotone and stable in $L^infty$, hence convergent by Barles-Perthame Souganidis.