Semi-groups and the mean reverting SABR stochastic volatility model

Abstract

We continue our study of solutions to linear parabolic partial differential equations (PDEs) by means of an asymptotic method that is based on approximate Green functions. A substantial part of this method is devoted to constructing the approximate Green function. In this paper, we approximate the Green function (or heat kernel) by asymptotically developing it in a small parameter other than time. While the method is general, in order to better illustrate it, we concentrate on the \(\lambda\)-SABR partial differential equation (PDE for short), which we study in detail. The \(\lambda\)-SABR PDE is a particular evolution PDE that arises in applications to stochastic volatility models (Hagan, Kumar, Lesniewski, and Woodward, Wilmott Magazine, 2002). Concretely, we study the generation and approximation of several semi-groups associated to the SABR PDE, some of which are non-standard because their generators are not uniformly elliptic and have unbounded coefficients. These type of generators appear also in the study of quasi-linear evolution equations. For some of the resulting semi-groups, we obtain explicit formulas by using a general technique based on solvable Lie groups that we develop in this paper. We thus obtain a simple, explicit approximation for the solution of the \(\lambda\)-SABR PDE and we prove explicit error bounds. In view of the potential applications, we have tried to make our paper as self-contained as reasonably possible.