Skip to main content
On refined volatility smile expansion in the Heston model
Quant. Finance
  • Peter K. Friz, TU Berlin
  • Stefan Gerhold, TU Wien
  • Archil Gulisashvili, Ohio University
  • Stephan Sturm, WPI, Department of Mathematical Sciences
Document Type
Publication Date
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 ST (equivalently the Mellin transform of ST ). 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.
Citation Information
Peter K. Friz, Stefan Gerhold, Archil Gulisashvili and Stephan Sturm. "On refined volatility smile expansion in the Heston model" Quant. Finance Vol. 11 Iss. 8 (2011) p. 1151 - 1164
Available at: