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: σBS(k, T)2 T ∼ Ψ(s + − 1) × 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 Drăgulescu and Yakovenko [Quant. Finance, 2002, 2(6), 443–453], and then show the validity of a refined expansion of the type σBS(k, T)2 T = (β1 k 1/2 + β2 + ···)2, where all constants are explicitly known as functions of s +, the Heston model parameters, the 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., 2010, 61(3), 287–315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles and at no point do we need knowledge of the (explicit, but cumbersome) closed-form expression of the Fourier transform of log S T (equivalently the Mellin transform of S T ). What matters is that these transforms satisfy ordinary differential equations of the Riccati type. Secondly, our analysis reveals a new parameter (the ‘critical slope’), defined in a model-free manner, which drives the second- and higher-order terms in tail and implied volatility expansions.