Pricing Multidimensional Financial Derivatives with Stochastic Volatilities Using the Dimensional-Adaptive Combination Technique

In this paper, we present a new and general approach to price derivatives based on the Black–Scholes partial differential equation (BS-PDE) in a multidimensional setting. The first ingredient in our approach is the dimensional-adaptive sparse grid combination technique, which, in the case of underlying models with stochastic volatilities, allows for inhomogeneous discretization levels of the dimensional axes. Thus, by applying the dimensional-adaptive combination technique to such problems, one may achieve higher numerical efficiency. We combine this approach with a stretched grid discretization that is derived from the underlying’s stochastic differential equation (SDE) in a general manner. This stretching enables us to employ efficient geometrical multigrid solvers, even for the strong anisotropic convection and diffusion coefficients that frequently occur in application. Our combination of the dimensional-adaptive sparse grid combination technique with SDE-based grid stretching and an efficient multigrid solver represents a new approach designed to enable derivative pricing by directly solving PDEs in higher dimensions than were possible before. The numerical results outlined in the paper demonstrate the efficacy of this new approach and of our implementation method, which entails pricing various derivatives with up to twelve dimensions in a general and simple manner.