Giles, Szpruch - Multi levelito MC
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a r X i v : 1 2 1 2 . 1 3 7 7 v 1 [ q f i n . C P ] 6 D e c 2 0 1 2 Multilevel Monte Carlo methods for applications in finance Mike Giles and Lukasz Szpruch Oxford-Man Institute of Quantitative Finance and Mathematical Institute, University of Oxford August 22, 2013 Abstract Since Giles introduced the multilevel Monte Carlo path simulation method [18], there has been rapid development of the technique for a variet y of applica tions in compu tational finance. This paper survey s the progress so far, highlights the key features in achieving a high rate of mult ilev el variance con ve rgence, and sugg ests direc tions for future research. 1 Introduct ion In 2001, Heinrich [28], developed a multilevel Monte Carlo method for para- metric inte gration, in which one is interested in estimating the value of E[f (x, λ)] where x is a finite-dimensional random variable and λ is a parameter. In the simplest case in which λ is a real variable in the range [0 , 1], having estimated the value of E[f (x, 0)] and E[f (x, 1)], one can use 1 2 (f (x, 0) + f (x, 1)) as a control variate when estimating the value of E[f (x, 1 2 )], since the variance of f (x, 1 2 ) − 1 2 (f (x, 0) + f (x, 1)) will usually be less than the variance of f (x, 1 2 ). This approach can then be applied recursively for other intermediate values of λ, yielding large savings if f (x, λ) is sufficiently smooth with respect to λ. Giles’ multilevel Monte Carlo path simulation [18] is both similar and dif- ferent. There is no parametric integration, and the random variable is infinite- dimensional, corresponding to a Brownian path in the original paper. However, the control variate viewpoint is very similar. A coarse path simulations is used as a control variate for a more refined fine path simulation, but since the ex- act expectation for the coarse path is not known, this is in turn estimated recursively using even coarser path simulation as control variates. The coars- est path in the multilevel hierarchy may have only one timestep for the entire interval of interest. 1
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