Moment explosions, implied volatility and local volatility at extreme strikes

TL;DR

This paper analyzes how the finiteness of the moment generating function in stochastic volatility models influences the asymptotic behavior of implied and local volatility at extreme strikes, providing precise estimates.

Contribution

It introduces a new Tauberian approach to derive sharp asymptotics of implied and local volatility from the behavior of the moment generating function near its critical point.

Findings

Asymptotic expansions for implied volatility at extreme strikes

Application to Gatheral's SVI parametrization

Application to Heston's model

Abstract

We consider a stochastic volatility model where the moment generating function of the logarithmic price is finite only on part of the real line. Using a new Tauberian result obtained in [1] and [2], we show that the knowledge of the moment generating function near its critical moment gives a sharp asymptotic expansion (with an error of order o(1)) of the local volatility and implied volatility for small and large strikes. We apply our theoretical estimates to Gatheral's SVI parametrization of the implied volatility and Heston's model.