Fast and efficient numerical methods for an extended Black–Scholes model

We study fast and efficient numerical approximations of a partial integro-differential equation (PIDE) that arises in option pricing theory (financial problems) as well as in various scientific modeling and deal with two different topics. As a first part of this article, we rigorously study different preconditioned iterative schemes for the full discrete model equation, focusing in particular on the choice of a fast and efficient solver. We develop preconditioners using wavelet basis transformations and Fourier sine transformations to improve the convergence criteria of iterative solvers for the full discrete model. We implement a multigrid (MG) iterative method for the discrete model problem as well. In fact, we approximate the problem using a finite difference scheme, then implement a few preconditioned Krylov subspace methods as well as a MG method to speed up the computation. A series of numerical experiments demonstrates that the number of iterations required for convergence is small and independent of the system size. In this sense the methods are optimal. Then, in the second part in this study, we analyze the stability and the accuracy of two different one step schemes to approximate the model.