Second Order Accurate IMEX Methods for Option Pricing Under Merton and Kou Jump-Diffusion Models

Abstract

In this paper three implicit-explicit (IMEX) time semi-discrete methods, namely IMEX-BDF1, IMEX-BDF2 and CN-LF, are developed for solving parabolic partial integro-differential equations which arise in option pricing theory when the underlying asset follows a jump diffusion process. It is shown that IMEX-BDF2 and CN-LF are stable and second order accurate, whereas IMEX-BDF1 is stable but only first order accurate. After time semi-discretization, the resulting linear differential equations are solved by using a cubic B-spline collocation method. The methods so developed have computational complexity of $$O(MNlog_{2}(M))$$O(MNlog2(M)) for Merton model and of $$O(MN)$$O(MN) for Kou model, where $$N$$N denotes the number of time steps and $$M$$M the number of collocation points. Some numerical examples, for pricing European options under Merton and Kou jump-diffusion models with constant as well as variable volatility, are presented to demonstrate the stability, convergence and computational complexity of the methods.