Economy – Quantitative Finance – Computational Finance
Scientific paper
2012-02-28
Economy
Quantitative Finance
Computational Finance
Scientific paper
In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multidimensional SDEs driven by Brownian motion. Giles has previously shown that if we combine a numerical approximation with strong order of convergence $O(\D t)$ with MLMC we can reduce the computational complexity to estimate expected values of functionals of SDE solutions with a root-mean-square error of $\eps$ from $O(\eps^{-3})$ to $O(\eps^{-2})$. However, in general, to obtain a rate of strong convergence higher than $O(\D t^{1/2})$ requires simulation, or approximation, of \Levy areas. In this paper, through the construction of a suitable antithetic multilevel correction estimator, we are able to avoid the simulation of \Levy areas and still achieve an $O(\D t^2)$ variance for smooth payoffs, and almost an $O(\D t^{3/2})$ variance for piecewise smooth payoffs, even though there is only $O(\D t^{1/2})$ strong convergence. This results in an $O(\eps^{-2})$ complexity for estimating the value of European and Asian put and call options.
Giles Michael B.
Szpruch Lukasz
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