Applications of Malliavin calculus to Monte-Carlo methods in finance. II
The valuation and the hedging of most financial products involve expected values, and their differentials, of functionals of Brownian motions and related stochastic processes. A general and natural approach to the practical (numerical) computation of such quantities is the Monte Carlo method. The advantages of Monte Carlo simulations on other numerical approaches are the flexibility and the possibility of dealing with path-dependent products, and the potential for computations in higher dimensions. The drawbacks of Monte Carlo approaches are: 1) the relative slow convergence and the lack of precision; 2) the inadequacy for the treatment of American options. The aim of this paper (as well as Part I of the paper by the same authors [ibid. 3, No. 4, 391--412 (1999; Zbl 0947.60066)]) is to provide some curves for these drawbacks. In Part I it was shown how one can use Malliavin calculus to write down explicit probabilistic formulas for the greeks -- i.e., differentials of the values of various European options with respect to various parameters -- which are needed for hedging. In this paper, several other applications of Malliavin calculus are presented. First of all, the question of the weights appearing in the representations of greeks mentioned above is analyzed: indeed (Part I), those weights are non unique and thus it needs to choose the ``best'' one. The use of Malliavin calculus to compute conditional expectations is investigated. The integration by parts formula provides a powerful tool when used in the framework of Monte Carlo simulation. It allows to compute everywhere, on a single set of trajectories starting at one point, solutions of general options related to PDEs. The final application of Malliavin calculus concerns the use of Girsanov transforms involving anticipating drifts.
scientific article; zbMATH DE number 1617473
scientific article