On a Symmetrization of Diffusion Processes

TL;DR

This paper provides a mathematical foundation for a symmetrization-based numerical scheme used to price barrier options in complex stochastic volatility models, extending its applicability to various diffusion processes and boundary conditions.

Contribution

It offers a general theoretical result on symmetrization for multi-dimensional diffusions, validating and broadening the scheme's use in financial modeling.

Findings

Mathematical validation of symmetrization scheme for Heston and SABR models

Extension of symmetrization to time-inhomogeneous and curved boundaries

Discussion of applications to diverse diffusion processes

Abstract

The latter author, together with collaborators, proposed a numerical scheme to calculate the price of barrier options. The scheme is based on a symmetrization of diffusion process. The present paper aims to give a mathematical credit to the use of the numerical scheme for Heston or SABR type stochastic volatility models. This will be done by showing a fairly general result on the symmetrization (in multi-dimension/multi-reflections). Further applications (to time-inhomogeneous diffusions/ to time dependent boundaries/to curved boundaries) are also discussed.