Weighted Monte Carlo with Least Squares and Randomized Extended Kaczmarz for Option Pricing

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.

The randomized circumcentered-reflection iteration method for solving consistent linear equations

“…where A has full column rank, which usually comes from the practical applications such as image reconstruction [10], phase retrieval [25], option pricing [8], biological computation [9] and machine learning [16].…”

On fast greedy block Kaczmarz methods for solving large consistent linear systems

“…where A ∈ R m×n and b ∈ R m , one of the classical and popular iteration methods is the Kaczmarz method [1]. Due to its simplicity and efficiency, it was deeply studied and widely used in many practical scientific and engineering applications, for instance, computer tomography(CT) [2], image reconstruction [3], machine learning [4] and option pricing [5].…”