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

By Ahmed Kebaier. 2005

140 citations recorded by the source corpus.

We study the approximation of Ef(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 delta and delta(beta), 0 < beta < 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 beta (which turns out to be beta = 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.

Subfields: Finance, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/54a513fc761b4a34fa5569133fb86d7294f9df3e

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Citations in the corpus

Improved Multilevel Monte Carlo Convergence using the Milstein Scheme (2008, methodology): “This multilevel approach is related to the two-level method of Kebaier [Keb05], and is similar to the multi-level method proposed by Speight [Spe05] based on the quasi control variate method of Emsermann and Simon [ES02].”
Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients (2013, background): “It is fruitful to consider the strong convergence because: pathwise convergence, relevant for assets models in mathematical finance, follows automatically [20]; efficient Monte Carlo simulations relay on both weak and strong convergence, for example, multilevel Monte Carlo methods [11,19].”
Multilevel Richardson-Romberg Extrapolation (2014, methodology): “…this paper is to combine the multilevel Monte Carlo estimator introduced by S. Heinrich in [Hei01] and developed by M. Giles in [Gil08] (see also [Keb05] for the statistical Romberg approach) and the (consistent) Multistep Richardson-Romberg extrapolation (see [Pag07]) in order to minimize the…”