# Variance Reduction Applied to Machine Learning for Pricing Bermudan/American Options in High Dimension

## TL;DR

This paper introduces a variance reduction technique combined with machine learning and Monte Carlo methods to efficiently price high-dimensional multi-asset American options, overcoming the curse of dimensionality.

## Contribution

It develops a novel algorithm that integrates control variates with Gaussian process regression for high-dimensional American option pricing.

## Key findings

- The method is fast and reliable for large baskets.
- Variance reduction improves accuracy in high dimensions.
- The approach outperforms traditional methods in high-dimensional settings.

## Abstract

In this paper we propose an efficient method to compute the price of multi-asset American options, based on Machine Learning, Monte Carlo simulations and variance reduction technique. Specifically, the options we consider are written on a basket of assets, each of them following a Black-Scholes dynamics. In the wake of Ludkovski's approach (2018), we implement here a backward dynamic programming algorithm which considers a finite number of uniformly distributed exercise dates. On these dates, the option value is computed as the maximum between the exercise value and the continuation value, which is obtained by means of Gaussian process regression technique and Monte Carlo simulations. Such a method performs well for low dimension baskets but it is not accurate for very high dimension baskets. In order to improve the dimension range, we employ the European option price as a control variate, which allows us to treat very large baskets and moreover to reduce the variance of price estimators. Numerical tests show that the proposed algorithm is fast and reliable, and it can handle also American options on very large baskets of assets, overcoming the problem of the curse of dimensionality.