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

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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)

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    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.

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    Bibliographic Info

    Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Journal of Theoretical and Applied Finance.

    Volume (Year): 15 (2012)
    Issue (Month): 04 ()
    Pages: 1250026-1-1250026-27

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    Handle: RePEc:wsi:ijtafx:v:15:y:2012:i:04:p:1250026-1-1250026-27

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    Related research

    Keywords: Implied volatility; term structure; symmetric smile; SVI; Heston; real-valued functions;

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