On refined volatility smile expansion in the Heston model

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.

References in corpus (1)

• Asymptotic formulae for implied volatility in the Heston model