Solving high-dimensional optimal stopping problems using optimization based model order reduction

TL;DR

This paper introduces a model order reduction technique for high-dimensional optimal stopping problems, enabling efficient Bermudan option pricing by reducing computational complexity and overcoming the curse of dimensionality.

Contribution

It develops a dimension reduction scheme based on an error measure for stochastic differential equations, facilitating feasible regression-based solutions in high dimensions.

Findings

Reduces computational complexity in high-dimensional optimal stopping problems.

Achieves accurate Bermudan option pricing in reduced models.

Demonstrates effectiveness through numerical experiments.

Abstract

Solving optimal stopping problems by backward induction in high dimensions is often very complex since the computation of conditional expectations is required. Typically, such computations are based on regression, a method that suffers from the curse of dimensionality. Therefore, the objective of this paper is to establish dimension reduction schemes for large-scale asset price models and to solve related optimal stopping problems (e.g. Bermudan option pricing) in the reduced setting, where regression is feasible. The proposed algorithm is based on an error measure between linear stochastic differential equations. We establish optimality conditions for this error measure with respect to the reduce system coefficients and propose a particular method that satisfies these conditions up to a small deviation. We illustrate the benefit of our approach in several numerical experiments, in which Bermudan option prices are determined.