On adaptive stratification

TL;DR

This paper introduces an adaptive stratified sampling method that dynamically adjusts partitions and sample allocations to significantly reduce variance in high-dimensional integral approximations, especially for complex financial models.

Contribution

It proposes a novel on-the-fly adaptive stratification technique that jointly optimizes partitioning and sampling, enhancing efficiency over traditional methods.

Findings

Variance reduction factors exceeding 1000 times compared to classical Monte Carlo.

Effective in high-dimensional, complex financial models with path-dependent options.

Demonstrates improved estimator accuracy in asymptotic regimes.

Abstract

This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the strata, which should be ideally fitted to thesubsets where the functions to integrate is nearly constant, and on the allocation of the number of samples within each strata. When the dimension is large and the function to integrate is complex, finding such partitions and allocating the sample is a highly non-trivial problem. In this work, we investigate a novel method to improve the efficiency of the estimator "on the fly", by jointly sampling and adapting the strata and the allocation within the strata. The accuracy of estimators when this method is used is examined in detail, in the so-called asymptotic regime (i.e. when both the number of samples and the number of strata are large). We illustrate the use of the method for the computation of the price of path-dependent options in models with both constant and stochastic volatility. The use of this adaptive technique yields variance reduction by factors sometimes larger than 1000 compared to classical Monte Carlo estimators.