On the Numerical Evaluation of Option Prices in Jump Diffusion Processes

The fair price of a financial option on an asset that follows a Poisson jump diffusion process satisfies a partial integro-differential equation. When numerical methods are used to solve such equations the integrals are usually evaluated using either quadrature methods or fast Fourier methods. Quadrature methods are expensive since the integrals must be evaluated at every point of the mesh. Though less so, Fourier methods are also computationally intensive since in order to avoid wrap around effects they require enlargement of the computational domain. They are also slow to converge when the parameters of the jump process are not smooth, and for efficiency require uniform meshes. We present a different and more efficient class of methods which are based on the fact that the integrals often satisfy differential equations. Depending on the process the asset follows, the equations are either ordinary differential equations or parabolic partial differential equations. Both types of equations can be accurately solved very rapidly. We discuss the methods and present results of numerical experiments.