Asymptotic Implied Volatility at the Second Order with Application to the SABR Model

TL;DR

This paper develops a method to compute higher-order asymptotic expansions of implied volatility in stochastic volatility models, providing explicit formulas up to second order, with an application to the SABR model.

Contribution

It introduces a heat kernel expansion approach to derive second-order implied volatility asymptotics at all strikes, extending previous first-order results.

Findings

Exact first-order correction at all strikes

Second-order correction derived explicitly

Application to SABR model demonstrated

Abstract

We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kernel expansion. Furthermore, the first correction in the heat kernel expansion gives the second order correction for implied volatility, which we also give exactly at all strikes. As an application, we compute this asymptotic expansion at order 2 for the SABR model.