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On refined volatility smile expansion in the Heston model


  • Peter Friz
  • Stefan Gerhold
  • Archil Gulisashvili
  • Stephan Sturm


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)2T ∼ 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)2T = (β1k1/2 + 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.

Suggested Citation

  • Peter Friz & Stefan Gerhold & Archil Gulisashvili & Stephan Sturm, 2011. "On refined volatility smile expansion in the Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1151-1164.
  • Handle: RePEc:taf:quantf:v:11:y:2011:i:8:p:1151-1164
    DOI: 10.1080/14697688.2010.541486

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    Cited by:

    1. Akihiko Takahashi, 2015. "Asymptotic Expansion Approach in Finance," CARF F-Series CARF-F-356, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.
    2. Archil Gulisashvili & Josep Vives, 2014. "Asymptotic analysis of stock price densities and implied volatilities in mixed stochastic models," Papers 1403.5302,
    3. Stefan Gerhold & Christoph Gerstenecker & Arpad Pinter, 2018. "Moment Explosions in the Rough Heston Model," Papers 1801.09458,, revised Apr 2018.
    4. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
    5. Sidi Mohamed Aly, 2016. "Moment explosions, implied volatility and local volatility at extreme strikes," Papers 1601.06995,, revised Aug 2016.
    6. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2017. "Shapes of implied volatility with positive mass at zero," Working Papers 2017-77, Center for Research in Economics and Statistics.


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