Stratified Monte Carlo simulation of Markov chains

TL;DR

This paper introduces stratified Monte Carlo methods for simulating Markov chains with continuous state spaces, demonstrating variance reduction and efficiency improvements in financial option pricing.

Contribution

It develops stratified sampling strategies for Markov chain simulation and analyzes their variance reduction benefits in financial applications.

Findings

Variance reduction in measure estimation using stratified samples

Enhanced efficiency in European and Asian option pricing

Parallel simulation of multiple Markov chains with sorted stratified samples

Abstract

We present several Monte Carlo strategies for simulating discrete-time Markov chains with continuous multi-dimensional state space; we focus on stratified techniques. We first analyze the variance of the calculation of the measure of a domain included in the unit hypercube, when stratified samples are used. We then show that each step of the simulation of a Markov chain can be reduced to the numerical integration of the indicator function of a subdomain of the unit hypercube. Our approach for Markov chains simulates N copies of the chain in parallel using stratified sampling and the copies are sorted after each step, according to their successive coordinates. We analyze variance reduction on examples of pricing of European and Asian options: enhanced efficiency of stratified strategies is shown.