A note on essential smoothness in the Heston model

TL;DR

This paper addresses an essential smoothness issue in the Heston model's large deviations application, correcting a proof gap and confirming the limiting implied volatility behavior far from maturity.

Contribution

It identifies and resolves a proof gap related to essential smoothness in the Heston model's large deviations analysis, completing the main result on implied volatility limits.

Findings

Resolved a proof gap in the Heston model analysis

Confirmed the limiting implied volatility behavior

Ensured the validity of the main theoretical result

Abstract

This note studies an issue relating to essential smoothness that can arise when the theory of large deviations is applied to a certain option pricing formula in the Heston model. The note identifies a gap, based on this issue, in the proof of Corollary 2.4 in \cite{FordeJacquier10} and describes how to circumvent it. This completes the proof of Corollary 2.4 in \cite{FordeJacquier10} and hence of the main result in \cite{FordeJacquier10}, which describes the limiting behaviour of the implied volatility smile in the Heston model far from maturity.