Time Dependent Heston Model

37 Pages Posted: 25 Mar 2009  

Eric Benhamou

Université Paris Est - Université Paris Est-Creteil

Emmanuel Gobet

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees

Mohammed Miri

Thomson Reuters

Date Written: March 24, 2009

Abstract

The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [Hes93] or piecewise constant [MN03]. Hence, using a small volatility of volatility expansion and Malliavin calculus techniques, we derive an accurate analytical formula for the price of vanilla options for any time dependent Heston model (the accuracy is less than a few bps for various strikes and maturities). In addition, we establish tight error estimates. The advantage of this approach over Fourier based methods is its rapidity (gain by a factor 100 or more), while maintaining a competitive accuracy. From the approximative formula, we also derive some corollaries related first to equivalent Heston models (extending some work of Piterbarg on stochastic volatility models [Pit05]) and second, to the calibration procedure in terms of ill-posed problems.

Keywords: asymptotic expansion, Malliavin calculus, small volatility of volatility, time dependent Heston model

Suggested Citation

Benhamou, Eric and Gobet, Emmanuel and Miri, Mohammed, Time Dependent Heston Model (March 24, 2009). Available at SSRN: https://ssrn.com/abstract=1367955 or http://dx.doi.org/10.2139/ssrn.1367955

Eric Benhamou

Université Paris Est - Université Paris Est-Creteil ( email )

61 avenue du Général de Gaulle
Créteil, 940000
France

Emmanuel Gobet (Contact Author)

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

Palaiseau Cedex, 91128
France

Mohammed Miri

Thomson Reuters ( email )

6 Bd Haussman
France, FL 75009
France

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