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November 2005 Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing
Ahmed Kebaier
Ann. Appl. Probab. 15(4): 2681-2705 (November 2005). DOI: 10.1214/105051605000000511

Abstract

We study the approximation of $\mathbb{E}f(X_{T})$ by a Monte Carlo algorithm, where X is the solution of a stochastic differential equation and f is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps δ and δβ,0<β<1. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal β (which turns out to be β=1/2), we establish a central limit type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error.

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Ahmed Kebaier. "Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing." Ann. Appl. Probab. 15 (4) 2681 - 2705, November 2005. https://doi.org/10.1214/105051605000000511

Information

Published: November 2005
First available in Project Euclid: 7 December 2005

zbMATH: 1099.65011
MathSciNet: MR2187308
Digital Object Identifier: 10.1214/105051605000000511

Subjects:
Primary: 60F05, 65C05
Secondary: 62P05

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.15 • No. 4 • November 2005
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