Open Access
August 2014 Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation
Michael B. Giles, Lukasz Szpruch
Ann. Appl. Probab. 24(4): 1585-1620 (August 2014). DOI: 10.1214/13-AAP957

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(ϵ3) to O(ϵ2). However, in general, to obtain a rate of strong convergence higher than O(Δt1/2) 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(ϵ2) complexity for estimating the value of European and Asian put and call options.

Citation

Download Citation

Michael B. Giles. Lukasz Szpruch. "Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation." Ann. Appl. Probab. 24 (4) 1585 - 1620, August 2014. https://doi.org/10.1214/13-AAP957

Information

Published: August 2014
First available in Project Euclid: 14 May 2014

zbMATH: 1373.65007
MathSciNet: MR3211005
Digital Object Identifier: 10.1214/13-AAP957

Subjects:
Primary: 65C05 , 65C30

Keywords: Lévy area , Monte Carlo , multilevel , Stochastic differential equation

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.24 • No. 4 • August 2014
Back to Top