Time Dependent Heston Model

37 Pages Posted: 25 Mar 2009

See all articles by Eric Benhamou

Eric Benhamou

Université Paris Dauphine; EB AI Advisory; AI For Alpha

Emmanuel Gobet

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees

Mohammed Miri

Thomson Reuters

Date Written: March 24, 2009


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 Dauphine ( email )

Place du Maréchal de Tassigny
Paris, Cedex 16 75775

EB AI Advisory ( email )

35 Boulevard d'Inkermann
Neuilly sur Seine, 92200

AI For Alpha ( email )

35 boulevard d'Inkermann
Neuilly sur Seine, 92200

Emmanuel Gobet (Contact Author)

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

Palaiseau Cedex, 91128

Mohammed Miri

Thomson Reuters ( email )

6 Bd Haussman
France, FL 75009

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