La simulation de Monte Carlo: forces et faiblesses (avec applications Visual Basic et Matlab et présentation d’une nouvelle méthode QMC)

Abstract

Monte Carlo simulation has an advantage upon the binomial tree as it can take into account the multidimensions of a problem. However it convergence speed is slower. In this article, we show how this method may be improved by various means: antithetic variables, control variates and low discrepancy sequences: Faure, Sobol and Halton sequences. We show how to compute the standard deviation of a Monte Carlo simulation when the payoffs of a claim, like a contingent claim, are nonlinear. In this case, we must compute this standard deviation by doing a great number of repeated simulations such that we arrive at a normal distribution of the results. The mean of the means of these simulations is then a good estimator of the wanted price. We also show how to combine Halton numbers with antithetic variables to improve the convergence of a QMC. That is our new version of QMC which is then well named because the result varies from one simulation to the other in our version of the QMC while the result is fixed (not random) in a classical QMC, like in the binomial tree.Financial engineering, derivatives, Monte Carlo simulation, low discrepancy sequences.