The Annals of Applied Probability

Integration by parts formula for locally smooth laws and applications to sensitivity computations

Vlad Bally, Marie-Pierre Bavouzet, and Marouen Messaoud

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Abstract

We consider random variables of the form F=f(V1, …, Vn), where f is a smooth function and Vi, i∈ℕ, are random variables with absolutely continuous law pi(y) dy. We assume that pi, i=1, …, n, are piecewise differentiable and we develop a differential calculus of Malliavin type based on ∂lnpi. This allows us to establish an integration by parts formula E(∂iϕ(F)G)=E(ϕ(F)Hi(F, G)), where Hi(F, G) is a random variable constructed using the differential operators acting on F and G. We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a Lévy process.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 1 (2007), 33-66.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1171377176

Digital Object Identifier
doi:10.1214/105051606000000592

Mathematical Reviews number (MathSciNet)
MR2292579

Zentralblatt MATH identifier
1139.60025

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60J75: Jump processes
Secondary: 65C05: Monte Carlo methods

Keywords
Malliavin calculus pure jump diffusions sensitivity analysis Monte Carlo algorithm European call and digital options

Citation

Bally, Vlad; Bavouzet, Marie-Pierre; Messaoud, Marouen. Integration by parts formula for locally smooth laws and applications to sensitivity computations. Ann. Appl. Probab. 17 (2007), no. 1, 33--66. doi:10.1214/105051606000000592. https://projecteuclid.org/euclid.aoap/1171377176


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References

  • Bavouzet, M. P. and Messaoud, M. (2006). Computation of Greeks using Malliavin calculus in jump type market models. Electron. J. Probab. 11 276–300.
    Mathematical Reviews (MathSciNet): MR2217817
    Zentralblatt MATH: 1113.60057
  • Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
    Mathematical Reviews (MathSciNet): MR1406564
  • Biagini, F., Øksendal, B., Sulem, A. and Wallner, N. (2004). An introduction to white noise and Malliavin calculus for fractional Brownian motion. Proc. Roy. Soc. Special Issue on Stochastic Analysis and Applications 460 347–372.
    Mathematical Reviews (MathSciNet): MR2052267
    Digital Object Identifier: doi:10.1098/rspa.2003.1246
    JSTOR: links.jstor.org
    Zentralblatt MATH: 1043.60044
  • Bichteler, K., Gravereaux, J. B. and Jacod, J. (1987). Malliavin Calculus for Processes with Jumps. Gordon and Breach, New York.
    Mathematical Reviews (MathSciNet): MR1008471
    Zentralblatt MATH: 0706.60057
  • Bouleau, N. (2003). Error Calculus for Finance and Physics: The Language of Dirichlet Forms. de Gruyter, Berlin.
    Mathematical Reviews (MathSciNet): MR2079474
  • Carlen, E. A. and Pardoux, É. (1990). Differential calculus and integration by parts on Poisson space. In Stochastics, Algebra and Analysis in Classical and Quantum Dynamics 63–73. Kluwer, Dordrecht.
    Mathematical Reviews (MathSciNet): MR1052702
    Zentralblatt MATH: 0685.60056
  • Cont, R. and Tankov, P. (2003). Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton, FL.
    Mathematical Reviews (MathSciNet): MR2042661
    Zentralblatt MATH: 1052.91043
  • Davis, M. H. A. and Johansson, M. (2006). Malliavin Monte Carlo Greeks for jump diffusions. Stochastic Process. Appl. 116 101–129.
    Mathematical Reviews (MathSciNet): MR2186841
    Digital Object Identifier: doi:10.1016/j.spa.2005.08.002
    Zentralblatt MATH: 1081.60040
  • Denis, L. (2000). A criterion of density for solutions of Poisson driven SDE's. Probab. Theory Related Fields 118 406–426.
    Mathematical Reviews (MathSciNet): MR1800539
    Digital Object Identifier: doi:10.1007/PL00008748
    Zentralblatt MATH: 0969.60064
  • Øksendal, B. (1996). An introduction to Malliavin calculus with applications to economics. Lecture notes, Norwegian School of Economics and Business Administration, Norway.
  • Forster, B., Lütkebohmert, E. and Teichmann, J. (2007). Calculation of Greeks for jump-diffusions. To appear.
  • Fournié, E., Lasry, J. M., Lebouchoux, J. and Lions, P. L. (2001). Applications of Malliavin calculus to Monte Carlo methods in finance II. Finance Stoch. 2 73–88.
  • Fournié, E., Lasry, J. M., Lebouchoux, J., Lions, P. L. and Touzi, N. (1999). Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 5 201–236.
  • Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland, Amsterdam.
    Mathematical Reviews (MathSciNet): MR1011252
    Zentralblatt MATH: 0684.60040
  • El Khatib, Y. and Privault, N. (2004). Computation of Greeks in a market with jumps via the Malliavin calculus. Finance Stoch. 8 161–179.
    Mathematical Reviews (MathSciNet): MR2048826
    Digital Object Identifier: doi:10.1007/s00780-003-0111-6
    Zentralblatt MATH: 1098.91050
  • Nualart, D. and Vives, J. (1990). Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités XXIV. Lecture Notes in Math. 1426 154–165. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1071538
    Zentralblatt MATH: 0701.60048
    Digital Object Identifier: doi:10.1007/BFb0083763
  • Di Nunno, G., Øksendal, B. and Proske, F. (2004). White noise analysis for Lévy processes. J. Funct. Anal. 206 109–148.
    Mathematical Reviews (MathSciNet): MR2024348
    Digital Object Identifier: doi:10.1016/S0022-1236(03)00184-8
    Zentralblatt MATH: 1078.60054
  • Picard, J. (1996). Formules de dualité sur l'espace de Poisson. Ann. Inst. H. Poincaré Probab. Statist. 32 509–548.
    Mathematical Reviews (MathSciNet): MR1411270
    Zentralblatt MATH: 0859.60045
  • Picard, J. (1996). On the existence of smooth densities for jump processes. Probab. Theory Related Fields 105 481–511.
    Mathematical Reviews (MathSciNet): MR1402654
    Zentralblatt MATH: 0853.60064
  • Privault, N. and Debelley, V. (2004). Sensitivity analysis of European options in the Merton model via the Malliavin calculus on the Wiener space. Preprint.
    Mathematical Reviews (MathSciNet): MR2112634
  • Privault, N. and Wei, X. (2004). A Malliavin calculus approach to sensitivity analysis in insurance. Insurance. Math. Econom. 35 679–690.
    Mathematical Reviews (MathSciNet): MR2106142
    Zentralblatt MATH: 1122.91044
  • Privault, N. and Wei, X. (2005). Integration by parts for point processes and Monte Carlo estimation. Preprint.
    Mathematical Reviews (MathSciNet): MR2355593
    Digital Object Identifier: doi:10.1239/jap/1189717546
    Project Euclid: euclid.jap/1189717546
    Zentralblatt MATH: 1128.62044
  • Vives, J., León, J. A., Utzet, F. and Solé, J. L. (2002). On Lévy processes, Malliavin calculus and market models with jumps. Finance Stoch. 6 197–225.
    Mathematical Reviews (MathSciNet): MR1897959
    Digital Object Identifier: doi:10.1007/s007800100055

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