Heat kernel methods in finance: the SABR model

TL;DR

This paper explores advanced heat kernel methods applied to the SABR model in finance, aiming to make complex geometric techniques accessible to finance professionals for improved stochastic volatility modeling.

Contribution

It introduces heat kernel expansion techniques from Riemannian geometry to the finance community, providing a more precise approximation for the SABR model.

Findings

Heat kernel methods yield more accurate SABR model approximations

The approach bridges complex geometry and financial modeling

Enhanced understanding of stochastic volatility dynamics

Abstract

The SABR model is a stochastic volatility model not admitting a closed form solution. Hagan, Kumar, Leniewski and Woodward have obtained an approximate solution by means of perturbative techniques. A more precise approximation was found by Henry-Labord\`ere with the heat kernel expansion method. The latter relies on deep and hard theorems from Riemannian geometry which are almost totally unknown to the professionals of finance, who however are those primarily interested in these results. The goal of this report is to fill this gap and to make these topics understandable with a basic knowledge of calculus and linear algebra.