THE TERM STRUCTURE OF IMPLIED VOLATILITY IN SYMMETRIC MODELS WITH APPLICATIONS TO HESTON

S. de Marco () and C. Martini ()
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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

International Journal of Theoretical and Applied Finance (IJTAF), 2012, vol. 15, issue 04, 1-27

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.

Keywords: Implied volatility; term structure; symmetric smile; SVI; Heston; real-valued functions (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (3)

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DOI: 10.1142/S0219024912500264

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