Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation

Abstract

In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multi-dimensional SDEs driven by Brownian motions. Giles has previously shown that if we combine a numerical approximation with strong order of convergence O(Δt) with MLMC we can reduce the computational complexity to estimate expected values of functionals of SDE solutions with a root-mean-square error of ε from O(ε-3) to O(ε-2). However, in general, to obtain a rate of strong convergence higher than O(Δt 1/2) requires simulation, or approximation, of Lévy areas. In this paper, through the construction of a suitable antithetic multilevel correction estimator, we are able to avoid the simulation of Lévy areas and still achieve an O(Δt2) multilevel correction variance for smooth payoffs, and almost an O(Δt3/2) variance for piecewise smooth payoffs, even though there is only O(Δt1/2) strong convergence. This results in an O(ε-2) complexity for estimating the value of European and Asian put and call options. © 2014 Institute of Mathematical Statistics.