Dirichlet Forms and Finite Element Methods for the SABR Model

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 nonstandard 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.The time derivative of a function u in the appropriate Bochner space is understood in the weak sense: For u ∈ L 2 (J; V), its weak derivative inu ∈ L 2 (J, V * ) ∩ H 1 (J; V * ) is defined by the relation (2.16) J (u(t), v) V * ×V ϕ(t)dt = − J (u(t), v) V * ×Vφ (t)dt, see [44, Sections 2.1 and 3.1] for definitions and properties of Bochner spaces.

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On a non-standard two-species stochastic competing system and a related degenerate parabolic equation

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“…High dimensional problems as in [19] are one useful applications of the speedup resuliting from this methodology. But it can also enable us to speed up more involved numerical methods for benchmark stochastic volatility models: multiple integrals [2], Monte Carlo-type methods [48] or Finite Element Methods [36] for the SABR model can thus compete in speed with the original SABR expansion formula [27], by pre-learning them through the DNN. In related contexts deep BSDE solvers have been used to replace Monte Carlo methods for solving Backward Stochastic Differential Equations in high dimension [28,33,49] which can arise from pricing problem.…”