Abstracts

In contemporary finance, partial differential equations (PDEs) with mixed spatial derivative terms play an important role in the pricing of option products. For the majority of these PDE problems no exact solution is available and one has to resort to numerical methods to find an approximate solution. Here, Alternating Direction Implicit (ADI) splitting schemes are very popular amongst practitioners as they are highly efficient in comparison to classical (non-splitted) methods. However, for practical applications, a structural analysis of their convergence properties is of paramount importance. To the best of our knowledge, no such analysis has been performed in the literature up to now for problems with mixed derivative terms. The Modified Craig-Sneyd (MCS) scheme is an often used ADI scheme, and in this talk we shall consider its convergence properties in the numerical solution of multi-dimensional convection-diffusion problems containing mixed spatial derivatives. We prove that, under some natural stability assumptions, the MCS scheme has an order of convergence equal to two uniformly in the spatial mesh width. Next, we apply the theory in the case of the Heston PDE which plays an important role in financial option pricing theory. Our numerical experiments point out that all the stability assumptions are fulfilled and we observe indeed a second-order convergence behavior that holds uniformly in the spatial mesh width.