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: 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 Dr \u{a}%DIFDELCMD < }%%% gulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expansion of the type \% BS( k,T) ^{2}T=( 1k^{1/2}+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. Opt. , DOI: 10.1007/s002450099085 ]. 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} \% ) . Secondly, our analysis reveals a new parameter (" \textit{critical slope " \% ), defined in a model free manner, which drives the second and higher order terms in tail- and implied volatility expansions.
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: 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 }%DIFDELCMD < }%%% Dragulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expansion of the type BS( k,T) ^{2}T=( 1k^{1/2}+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 (" \textit{critical slope " ), defined in a model free manner, which drives the second and higher order terms in tail- and implied volatility expansions.
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