Dirichlet Forms and Finite Element Methods for the SABR Model

TL;DR

This paper develops a novel finite element method using Dirichlet forms to accurately price vanilla options under the SABR model, effectively handling degeneracies at the origin and applicable in various interest rate environments.

Contribution

It introduces a new analytic framework and finite element discretization for SABR pricing equations, addressing degeneracy issues at the origin with weighted Sobolev spaces.

Findings

Well-posed variational formulation for SABR pricing equations.

Finite element scheme with error analysis based on wavelet approximation.

Method applicable in both moderate and near-zero interest rate regimes.

Abstract

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 $\theta$-scheme in time and provide an error analysis for this discretization.