The Annals of Applied Probability

Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing

Ahmed Kebaier

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 4 (2005), 2681-2705.

Dates
First available in Project Euclid: 7 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1133965776

Digital Object Identifier
doi:10.1214/105051605000000511

Mathematical Reviews number (MathSciNet)
MR2187308

Zentralblatt MATH identifier
1099.65011

Subjects
Primary: 65C05: Monte Carlo methods 60F05: Central limit and other weak theorems
Secondary: 62P05: Applications to actuarial sciences and financial mathematics

Keywords
Romberg extrapolation Monte Carlo simulation variance reduction central limit theorem option pricing finance

Citation

Kebaier, Ahmed. Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15 (2005), no. 4, 2681--2705. doi:10.1214/105051605000000511. https://projecteuclid.org/euclid.aoap/1133965776


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