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

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 $O(Δt^2)$ multilevel correction variance for smooth payoffs, and almost an $O(Δt^{3/2})$ variance for piecewise smooth payoffs, even though there is only $O(Δt^{1/2})$ strong convergence. This results in an $O(ε^{-2})$ complexity for estimating the value of European and Asian put and call options.

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