Antithetic Multilevel Monte Carlo Estimation for Multidimensional SDEs

Abstract

In this paper we develop antithetic multilevel Monte Carlo (MLMC) estimators for multidimensional SDEs driven by Brownian motion. Giles has previously shown that if we combine a numerical approximation with strong order of convergence O(Δ t) with MLMC we can reduce the computational complexity to estimate expected values of Lipschitz 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 thnan O(Δ t ^1∕2) requires simulation, or approximation, of Lévy areas. Recently, Giles and Szpruch [5] constructed an antithetic multilevel estimator thnnat avoids thnne simulation of Lévy areas and still achieves an MLMC correction variance which is O(Δ t ²) for smooth payoffs and almost O(Δ t ^3∕2) for piecewise smooth payoffs, even though there is only O(Δ t ^1∕2) strong convergence. This results in an O(ε ⁻²) complexity for estimating the value of financial European and Asian put and call options. In this paper, we extend these results to more complex payoffs based on the path minimum. To achieve this, an approximation of the Lévy areas is needed, resulting in O(Δ t ^3∕4) strong convergence. By modifying the antithetic MLMC estimator we are able to obtain O(ε ⁻²log(ε)²) complexity for estimating financial barrier and lookback options.