Dirichlet forms and finite element methods for the SABR Model

We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via nonsymmetric Dirichlet forms. Our pricing method is valid both in moderate interest rate environments and in low interest rate regimes, such as the currently prevalent ones, and is applicable under mild assumptions on parameter configurations of the process. The parabolic Kolmogorov pricing equations for the SABR model are degenerate at the origin, yielding nonstandard partial differential equations, for which conventional pricing methods—designed for nondegenerate parabolic equations—potentially break down. We derive here the appropriate analytic setup to handle the degeneracy of the model at the origin. That is, we construct an evolution triple of suitably chosen Sobolev spaces with singular weights, consisting of the domain of the SABR-Dirichlet form—its dual space—and the pivotal Hilbert space. In particular, we show well-posedness of the variational formulation of the SABR-pricing equations for vanilla and barrier options on this triple. Furthermore, we present finite element discretization scheme based on a (weighted) multiresolution wavelet approximation in space and a ?-scheme in time and provide an error analysis for the discretization.