On refined volatility smile expansion in the Heston model

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)T-2 similar to Psi(s(+) - 1) x 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, 2002, 2(6), 443-453], and then show the validity of a refined expansion of the type sigma(BS)(k, T)T-2 (beta(1)k(1/2) + beta(2)+ ...)(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 ST (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.