On the rates of convergence of simulation based optimization algorithms for optimal stopping problems

TL;DR

This paper analyzes the convergence rates of simulation-based algorithms for optimal stopping problems, providing theoretical bounds and practical guidance for their implementation in finance.

Contribution

It derives optimal convergence rates using large deviation theory, showing these rates cannot generally be improved, and offers practical insights for algorithm design.

Findings

Derived convergence rates for simulation algorithms

Showed rates are generally optimal and unimprovable

Provided numerical example in option pricing context

Abstract

In this paper we study simulation based optimization algorithms for solving discrete time optimal stopping problems. This type of algorithms became popular among practioneers working in the area of quantitative finance. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates and show that they can not be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in option pricing that illustrates our theoretical findings.