Solving High-Dimensional Optimal Stopping Problems Using Optimization Based Model Order Reduction

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 reduced 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.

Keywords:

• Asset price model
• option pricing in high dimensions
• optimization based model order reduction
• Lévy processes

Acknowledgment

This work is supported by the DFG via the individual grant “Low-order approximations for large-scale problems arising in the context of high-dimensional PDEs and spatially discretized SPDEs”.

Disclosure Statement

No potential conflict of interest was reported by the author.

Notes

1 $({\mathcal{F}}_t{)}_{t\in [0,T]}$ is right continuous and complete.