On refined volatility smile expansion in the Heston model

TL;DR

This paper refines the understanding of the implied volatility smile in the Heston model by deriving a new tail expansion for the density and a precise asymptotic expansion for implied volatility at large strikes, with explicit constants.

Contribution

It introduces a novel tail expansion for the Heston density and a refined asymptotic expansion for implied volatility, explicitly linking constants to model parameters and the critical slope.

Findings

Derived a sharper tail expansion for the Heston density.

Established a refined large-strike implied volatility expansion with explicit constants.

Identified a new model-free parameter called the 'critical slope'.

Abstract

It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment $s_+$ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: $\sigma_{BS}( k,T)^{2}T\sim \Psi (s_+-1) \times k$ (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Dragulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expansion of the type $\sigma_{BS}( k,T) ^{2}T=( \beta_{1}k^{1/2}+\beta_{2}+...)^{2}$, where all constants are explicitly known as functions of $s_+$, the Heston model parameters, spot vol and maturity $T$. In the case of the "zero-correlation" Heston model such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim. 61, 3 (2010), 287--315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles: at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of $\log S_{T}$\ (equivalently: Mellin transform of $S_{T}$ ); what matters is that these transforms satisfy ordinary differential equations of Riccati type. Secondly, our analysis reveals a new parameter ("critical slope"), defined in a model free manner, which drives the second and higher order terms in tail- and implied volatility expansions.