The Term Structure of Implied Volatility in Symmetric Models with Applications to Heston

25 Pages Posted: 13 Jun 2010 Last revised: 20 Nov 2010

See all articles by Stefano De Marco

Stefano De Marco

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees

Claude Martini

Zeliade Systems

Date Written: June 11, 2010

Abstract

We study the term structure of the implied volatility in a situation where the smile is symmetric. Starting from the result by Tehranchi 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 quadratic variation of the log price. To deal with the remaining part of the volatility surface, we build a time dependent SVI-type approximation which matches the ATM and extreme moneyness structure. As an instance of a symmetric model, we consider uncorrelated Heston: in this framework, our representation of the ATM volatility takes semiclosed (and easy to implement) form and the time-dependent SVI approximation displays considerable performances in a wide range of maturities and strikes. In addition, we show how to apply our results to a skewed smile by considering a displaced model. Finally, a noteworthy fact is that all along the paper we will deal only with Laplace transforms and not with Fourier transforms, thus avoiding any complex-valued function.

Keywords: Implied Volatility, Term Structure, Symmetric Smiles, SVI, Heston, Real-Valued Functions

JEL Classification: G13, C60, C63

Suggested Citation

De Marco, Stefano and Martini, Claude, The Term Structure of Implied Volatility in Symmetric Models with Applications to Heston (June 11, 2010). Available at SSRN: https://ssrn.com/abstract=1622828 or http://dx.doi.org/10.2139/ssrn.1622828

Stefano De Marco (Contact Author)

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees ( email )

Palaiseau Cedex, 91128
France

Claude Martini

Zeliade Systems ( email )

Paris
France

HOME PAGE: http://www.zeliade.com

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
479
Abstract Views
2,408
rank
75,141
PlumX Metrics