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

Abstract

International audienceWe 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

Multilevel Monte Carlo Path Simulation

“…The method extends the recent work of Kebaier [14] who proved that the computational cost of the simple problem described above can be reduced to O( −2.5 ) through the appropriate combination of results obtained using two levels of timestep, h and O(h 1/2 ). This is closely related to a more generally applicable approach of quasi control variates analysed by Emsermann and Simon [5].…”

Importance sampling and statistical Romberg method for Lévy processes

“…Moreover, using once again relation (18) we deduce the uniform boundedness of the family (κ ε ) ε>0 on the compact subset K. Hence, combining all these results together with assumption (WE υε ), we deduce the existence of c 3 > 0 not depending on ε such that…”

“…On the other hand, thanks to relation (18) we have the uniform equicontinuity of the family (κ ε ) ε>0 on the compact subset K. So, we only need to check this last property for the family (γ ε ) ε>0 in view to use after that Lemma 8.1 and then deduce the convergence of…”

“…Therefore, if we denote by K(ε) the cost of a single simulation of L ε T , then the total time complexity necessary to achieve the precision σ(ε) is given by C M C = O(K(ε)σ −2 (ε)) (see subsection 3.3). In order to improve the performance of this method we use the idea of the statistical Romberg method introduced by Kebaier [18] in the setting of Euler Monte Carlo methods for stochastic differential equations driven by a standard Brownian Motion which is also related to the well known Romberg's method introduced by Talay and Tubaro in [28]. Inspired by this technique, we introduce a novel method for the computation of our initial target.…”

Recent advances in various fields of numerical probability

“…We propose and analyze a new estimator called Multilevel Richardson-Romberg (ML2R) which combines the higher order bias cancellation of the Multistep Richardson-Romberg method introduced in [61] and the variance control resulting from the stratification in the Multilevel Monte Carlo (MLMC) method introduced in [29] (see also [37,43]). The Multilevel Monte Carlo method has been extensively applied to various fields of numerical probability recently.…”