ABSTRACT
In the perspective of numerics, the Feynman-Kac theorem that we recalled in Chapter 1 allows us to switch from finite difference scheme methods to Monte Carlo implementations. Monte Carlo is more efficient with a high number of underlyings, because it hardly depends on the dimension of the problem while the computational time of finite difference methods grows exponentially with the number of variables. In this chapter, we briefly recall the principle of the Monte Carlo method and focus on various issues related to it, such as reduction of variance, bias of discretization schemes, and Romberg’s extrapolation. In particular, we present new advanced results regarding the convergence of the Euler scheme, probably the most widely used discretization scheme, showing that the well known result for the rate of convergence, which assumes smooth payoffs, actually holds even for extremely irregular payoffs. We also devote a large section to the so-called Romberg extrapolation, a very simple technique for building high order discretization schemes from low order ones at almost no extra cost.
