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The Term Structure Of Implied Volatility In Symmetric Models With Applications To Heston

Author

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  • S. DE MARCO

    (Université Paris-Est - CERMICS, 6 et 8 avenue Blaise Pascal, 77455, Marne la Vallee Cedex 2, France)

  • C. MARTINI

    (Zeliade Systems, 56 Rue Jean-Jacques Rousseau, 75001 Paris, France)

Abstract

We study the term structure of the implied volatility in the presence of a symmetric smile. Exploiting the result by Tehranchi (2009) that a symmetric smile generated by a continuous martingale necessarily comes from a mixture of normal distributions, we derive representation formulae for the at-the-money (ATM) implied volatility level and curvature in a general symmetric model. As a result, the ATM curve is directly related to the Laplace transform of the realized variance. The representation formulae for the implied volatility and its curvature take semi-closed form as soon as this Laplace transform is known explicitly. To deal with the rest of the volatility surface, we build a time dependent SVI-type (Gatheral, 2004) model which matches the ATM and extreme moneyness structure. As an instance of a symmetric model, we consider uncorrelated Heston: in this framework, the SVI approximation displays considerable performances in a wide range of maturities and strikes. All these results can be applied to skewed smiles by considering a displaced model. Finally, a noteworthy fact is that all along the paper we avoid dealing with any complex-valued function.

Suggested Citation

  • S. De Marco & C. Martini, 2012. "The Term Structure Of Implied Volatility In Symmetric Models With Applications To Heston," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(04), pages 1-27.
    • Handle: RePEc:wsi:ijtafx:v:15:y:2012:i:04:n:s0219024912500264
    DOI: 10.1142/S0219024912500264
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    References listed on IDEAS

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    1. Alexey MEDVEDEV & Olivier SCAILLET, 2004. "A Simple Calibration Procedure of Stochastic Volatility Models with Jumps by Short Term Asymptotics," FAME Research Paper Series rp93, International Center for Financial Asset Management and Engineering.
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    Cited by:

    1. Jan Pospíšil & Tomáš Sobotka & Philipp Ziegler, 2019. "Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure," Empirical Economics, Springer, vol. 57(6), pages 1935-1958, December.
    2. Antoine Jacquier & Fangwei Shi, 2016. "The randomised Heston model," Papers 1608.07158, arXiv.org, revised Dec 2018.
    3. Jan Posp'iv{s}il & Tom'av{s} Sobotka & Philipp Ziegler, 2019. "Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure," Papers 1912.06709, arXiv.org.

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