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ANTITHETIC MULTILEVEL MONTE CARLO ESTIMATION FOR MULTI-DIMENSIONAL SDES WITHOUT LÉVY AREA SIMULATION
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ANTITHETIC MULTILEVEL MONTE CARLO ESTIMATION FOR MULTI-DIMENSIONAL SDES WITHOUT LÉVY AREA SIMULATION
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Journal Article
ANTITHETIC MULTILEVEL MONTE CARLO ESTIMATION FOR MULTI-DIMENSIONAL SDES WITHOUT LÉVY AREA SIMULATION
2014
Overview
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 (△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 ∊ from O(∊⁻³) to O(∊⁻²). However, in general, to obtain a rate of strong convergence higher than O (△t½) 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 (△t²) multilevel correction variance for smooth payoffs, and almost an O (△t3/2) variance for piecewise smooth payoffs, even though there is only O(∊t1/2) strong convergence. This results in an O(∊⁻²) complexity for estimating the value of European and Asian put and call options.
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