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Numerical computation of Theta in a jump-diffusion model by integration by parts

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  • Delphine David
  • Nicolas Privault

Abstract

Using the Malliavin calculus in time inhomogeneous jump-diffusion models, we obtain an expression for the sensitivity Theta of an option price (with respect to maturity) as the expectation of the option payoff multiplied by a stochastic weight. This expression is used to design efficient numerical algorithms that are compared with traditional finite-difference methods for the computation of Theta. Our proof can be viewed as a generalization of Dupire's integration by parts to arbitrary and possibly non-smooth payoff functions. In the time homogeneous case, Theta admits an expression from the Black-Scholes PDE in terms of Delta and Gamma but the representation formula obtained in this way is different from ours. Numerical simulations are presented in order to compare the efficiency of the finite-difference and Malliavin methods.

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  • Delphine David & Nicolas Privault, 2009. "Numerical computation of Theta in a jump-diffusion model by integration by parts," Quantitative Finance, Taylor & Francis Journals, vol. 9(6), pages 727-735.
    • Handle: RePEc:taf:quantf:v:9:y:2009:i:6:p:727-735
    DOI: 10.1080/14697680902814191
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    References listed on IDEAS

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    1. Davis, Mark H.A. & Johansson, Martin P., 2006. "Malliavin Monte Carlo Greeks for jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 101-129, January.
    2. Eric Benhamou, 2002. "Smart Monte Carlo: various tricks using Malliavin calculus," Quantitative Finance, Taylor & Francis Journals, vol. 2(5), pages 329-336.
    3. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
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