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

Full-text: Open access


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

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

First available in Project Euclid: 13 February 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • Bavouzet, M. P. and Messaoud, M. (2006). Computation of Greeks using Malliavin calculus in jump type market models. Electron. J. Probab. 11 276–300.
  • Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • 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.
  • Bichteler, K., Gravereaux, J. B. and Jacod, J. (1987). Malliavin Calculus for Processes with Jumps. Gordon and Breach, New York.
  • Bouleau, N. (2003). Error Calculus for Finance and Physics: The Language of Dirichlet Forms. de Gruyter, Berlin.
  • 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.
  • Cont, R. and Tankov, P. (2003). Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton, FL.
  • Davis, M. H. A. and Johansson, M. (2006). Malliavin Monte Carlo Greeks for jump diffusions. Stochastic Process. Appl. 116 101–129.
  • Denis, L. (2000). A criterion of density for solutions of Poisson driven SDE's. Probab. Theory Related Fields 118 406–426.
  • 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.
  • El Khatib, Y. and Privault, N. (2004). Computation of Greeks in a market with jumps via the Malliavin calculus. Finance Stoch. 8 161–179.
  • 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.
  • Di Nunno, G., Øksendal, B. and Proske, F. (2004). White noise analysis for Lévy processes. J. Funct. Anal. 206 109–148.
  • Picard, J. (1996). Formules de dualité sur l'espace de Poisson. Ann. Inst. H. Poincaré Probab. Statist. 32 509–548.
  • Picard, J. (1996). On the existence of smooth densities for jump processes. Probab. Theory Related Fields 105 481–511.
  • Privault, N. and Debelley, V. (2004). Sensitivity analysis of European options in the Merton model via the Malliavin calculus on the Wiener space. Preprint.
  • Privault, N. and Wei, X. (2004). A Malliavin calculus approach to sensitivity analysis in insurance. Insurance. Math. Econom. 35 679–690.
  • Privault, N. and Wei, X. (2005). Integration by parts for point processes and Monte Carlo estimation. Preprint.
  • 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.