Spline cubatures for expectations of diffusion processes and optimal stopping in higher dimensions (with computational finance in view)

Abstract

We develop certain cubature (quadrature) rules for expectations of diffusion processes in ℝN that are analogous to the well known spline interpolation quadratures for ordinary integrals. By incorporating such rules in appropriate backward induction procedures, we develop new numerical algorithms for solving free-boundary (optimal stopping) problems, or ordinary fixed-boundary problems. The algorithms developed in the paper are directly applicable to pricing contingent claims of both American and European types on multiple underlying assets. © 2008 Springer-Verlag Berlin Heidelberg.