Smart Monte Carlo: various tricks using Malliavin calculus. (English) Zbl 1405.91688
Summary: Current Monte Carlo pricing engines may face a computational challenge for the Greeks, not only because of their time consumption but also their poor convergence when using a finite difference estimate with a brute force perturbation. The same story may apply to conditional expectation. In this short paper, following E. Fournié et al. [Finance Stoch. 3, No. 4, 391–412 (1999; Zbl 0947.60066)], we explain how to tackle this issue using Malliavin calculus to smoothen the payoff to estimate. We discuss the relationship with the likelihood ratio method of M. Broadie and P. Glasserman [Manage. Sci. 42, No. 2, 269–285 (1996; Zbl 0881.90018)]. We show by numerical results the efficiency of this method and discuss when it is appropriate or not to use it. We see how to apply this method to the Heston model.
MSC:
| 91G60 | Numerical methods (including Monte Carlo methods) |
| 65C05 | Monte Carlo methods |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 91B70 | Stochastic models in economics |
References:
| [1] | Benhamou, E. 2000a. An Application of Malliavin Calculus to Continuous Time Asian Options, London School of Economics working paper. http://www.ericbenhamou.fr.st/documents/articles/Malliavin-Asian.pdf |
| [2] | Benhamou, E. 2000b. “Application of Malliavin calculus and Wiener chaos to option pricing theory”. In PhD Thesis, London School of Economics. http://www.ericbenhamou.fr.st/documents/articles/Thesis.pdf |
| [3] | Benhamou, E. 2000c. “Optimal Malliavin weighting function for the computation of the Greeks”. In Proc. Monte Carlo Congr (Monte Carlo, 2000) Math. Finance, at press http://www.ericbenhamou.fr.st/documents/articles/MF_OptimalMalliavin.pdf · Zbl 1049.91058 |
| [4] | Broadie, M and Glasserman, P. 1996. Estimating security price derivatives using simulation. Manag. Sci., 42: 269-85. · Zbl 0881.90018 |
| [5] | Fournié, E, Lasry, J M, Lebuchoux, J, Lions, P L and Touzi, N. 1999. Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stochastics, 3: 391-412. · Zbl 0947.60066 |
| [6] | Fournié, E, Lasry, J M, Lebuchoux, J and Lions, P L. 2001. Applications of Malliavin calculus to Monte Carlo methods in finance. II. Finance Stochastics, 5: 201-36. · Zbl 0973.60061 |
| [7] | Gobet, E and Kohatsu-Higa, A. 2001. Computation of Greeks for Barrier and Lookback Options Using Malliavin Calculus, Ecole Polytechnique, CMAP, R.I. N464 http://www.cmap.polytechnique.fr/_gobet/paper/RI464.ps · Zbl 1061.60054 |
| [8] | Lions, P L and Régnier, H. Monte Carlo computations of American options via Malliavin calculus. Monte Carlo., 2000, Conf.mimeo. |
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