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On refined volatility smile expansion in the Heston model

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  • P. Friz
  • S. Gerhold
  • A. Gulisashvili
  • S. Sturm
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    Abstract

    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: $\sigma_{BS}( k,T)^{2}T\sim \Psi (s_+-1) \times k$ (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 Dragulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expansion of the type $\sigma_{BS}( k,T) ^{2}T=( \beta_{1}k^{1/2}+\beta_{2}+...)^{2}$, 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 ("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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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1001.3003.

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    Date of creation: Jan 2010
    Date of revision: Nov 2010
    Handle: RePEc:arx:papers:1001.3003

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    1. Adrian A. Dragulescu & Victor M. Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Papers, arXiv.org cond-mat/0203046, arXiv.org, revised Nov 2002.
    2. A. Gulisashvili, 2009. "Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes," Papers, arXiv.org 0906.0394, arXiv.org.
    3. A. Dragulescu & V. M. Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Computing in Economics and Finance 2002, Society for Computational Economics 127, Society for Computational Economics.
    4. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, Springer, vol. 11(1), pages 29-50, January.
    5. Martin Forde & Antoine Jacquier & Aleksandar Mijatovic, 2009. "Asymptotic formulae for implied volatility in the Heston model," Papers, arXiv.org 0911.2992, arXiv.org, revised May 2010.
    6. Adrian Dragulescu & Victor Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(6), pages 443-453.
    7. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 19(1), pages 1-12.
    8. A. Gulisashvili & E. M. Stein, 2009. "Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models," Papers, arXiv.org 0906.0392, arXiv.org.
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