Weighted Monte Carlo with least squares and randomized extended Kaczmarz for option pricing

TL;DR

This paper introduces a novel Monte Carlo-based method combining weighted sampling, least squares, and randomized Kaczmarz algorithms to efficiently compute high-dimensional option prices and expectations, addressing memory and computational challenges.

Contribution

It generalizes a multivariate integration method to probability measures and develops an efficient large-scale least-squares solution for high-dimensional problems.

Findings

Effective in low and high dimensions

Reduces memory requirements in large-scale problems

Demonstrates convergence and efficiency through numerical experiments

Abstract

We propose a methodology for computing single and multi-asset European option prices, and more generally expectations of scalar functions of (multivariate) random variables. This new approach combines the ability of Monte Carlo simulation to handle high-dimensional problems with the efficiency of function approximation. Specifically, we first generalize the recently developed method for multivariate integration in [arXiv:1806.05492] to integration with respect to probability measures. The method is based on the principle "approximate and integrate" in three steps i) sample the integrand at points in the integration domain, ii) approximate the integrand by solving a least-squares problem, iii) integrate the approximate function. In high-dimensional applications we face memory limitations due to large storage requirements in step ii). Combining weighted sampling and the randomized extended Kaczmarz algorithm we obtain a new efficient approach to solve large-scale least-squares problems. Our convergence and cost analysis along with numerical experiments show the effectiveness of the method in both low and high dimensions, and under the assumption of a limited number of available simulations.