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 non-symmetric Dirichlet forms. Our
pricing method is valid under mild assumptions on parameter configurations of
the process both in moderate interest rate environments and in near-zero
interest rate regimes such as the currently prevalent ones. The parabolic
Kolmogorov pricing equations for the SABR model are degenerate at the origin,
yielding non-standard partial differential equations, for which conventional
pricing methods ---designed for non-degenerate 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 a finite element discretization scheme based on a
(weighted) multiresolution wavelet approximation in space and a θ-scheme
in time and provide an error analysis for this discretization