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: σBS(k, T ) 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 2, 6 (2002), 443–453], and then show the validity of a refined expansion of the type σBS(k, T ) T = (β1k 1/2 +β2+ . . . ) , 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 ST (equivalently: Mellin transform of ST ); 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 tailand implied volatility expansions.