Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR Stochastic Volatility Model

TL;DR

This paper employs Lie group techniques to analyze and approximate solutions for PDEs in stochastic volatility models, specifically the SABR model with mean reversion, providing new formulas and perturbation insights.

Contribution

It introduces an exact solution formula for a degenerate elliptic operator in the SABR model and develops a perturbation approach for the associated PDEs using Lie algebra methods.

Findings

Derived an exact formula for the solution operator of a degenerate elliptic PDE.

Established a perturbation framework comparing operators with zero and non-zero volvol.

Provided insights for numerical methods in stochastic volatility modeling.

Abstract

We use commutator techniques and calculations in solvable Lie groups to investigate certain evolution Partial Differential Equations (PDEs for short) that arise in the study of stochastic volatility models for pricing contingent claims on risky assets. In particular, by restricting to domains of bounded volatility, we establish the existence of the semi-groups generated by the spatial part of the operators in these models, concentrating on those arising in the so-called "SABR stochastic volatility model with mean reversion." The main goal of this work is to approximate the solutions of the Cauchy problem for the SABR PDE with mean reversion, a parabolic problem the generator of which is denoted by $L$. The fundamental solution for this problem is not known in closed form. We obtain an approximate solution by performing an expansion in the so-called volvol or volatility of the volatility, which leads us to study a degenerate elliptic operator $L_0$, corresponding the the zero-volvol case of the SABR model with mean reversion, to which the classical results do not apply. However, using Lie algebra techniques we are able to derive an exact formula for the solution operator of the PDE $\partial_t u - L_0 u = 0$. We then compare the semi-group generated by $L$--the existence of which does follows from standard arguments--to that generated by $L_0$, thus establishing a perturbation result that is useful for numerical methods for the SABR PDE with mean reversion. In the process, we are led to study semigroups arising from both a strongly parabolic and a hyperbolic problem.