Moving average options: Machine Learning and Gauss-Hermite quadrature for a double non-Markovian problem

TL;DR

This paper introduces GPR-GHQ, an efficient machine learning-based method for pricing complex moving average options with long windows and high-dimensional features, applicable to various financial models including rough volatility.

Contribution

The paper develops a novel GPR-GHQ approach combining Gaussian Process Regression and Gauss-Hermite quadrature for high-dimensional moving average option pricing, including non-Markovian models.

Findings

GPR-GHQ achieves high accuracy in complex models.

Method effectively handles long window, high-dimensional problems.

Outperforms traditional methods in computational efficiency.

Abstract

Evaluating moving average options is a tough computational challenge for the energy and commodity market as the payoff of the option depends on the prices of a certain underlying observed on a moving window so, when a long window is considered, the pricing problem becomes high dimensional. We present an efficient method for pricing Bermudan style moving average options, based on Gaussian Process Regression and Gauss-Hermite quadrature, thus named GPR-GHQ. Specifically, the proposed algorithm proceeds backward in time and, at each time-step, the continuation value is computed only in a few points by using Gauss-Hermite quadrature, and then it is learned through Gaussian Process Regression. We test the proposed approach in the Black-Scholes model, where the GPR-GHQ method is made even more efficient by exploiting the positive homogeneity of the continuation value, which allows one to reduce the problem size. Positive homogeneity is also exploited to develop a binomial Markov chain, which is able to deal efficiently with medium-long windows. Secondly, we test GPR-GHQ in the Clewlow-Strickland model, the reference framework for modeling prices of energy commodities. Finally, we consider a challenging problem which involves double non-Markovian feature, that is the rough-Bergomi model. In this case, the pricing problem is even harder since the whole history of the volatility process impacts the future distribution of the process. The manuscript includes a numerical investigation, which displays that GPR-GHQ is very accurate and it is able to handle options with a very long window, thus overcoming the problem of high dimensionality.