Importance sampling for option pricing with feedforward neural networks

TL;DR

This paper introduces a neural network-based approach to variance reduction in Monte Carlo option pricing, demonstrating that neural networks can approximate optimal sampling measures effectively, leading to improved pricing accuracy.

Contribution

It proves neural networks can universally approximate optimal sampling measures in Gaussian settings, enabling more efficient Monte Carlo estimations for complex option pricing models.

Findings

Neural networks can approximate optimal sampling measures arbitrarily well.

The proposed method reduces variance in Monte Carlo estimators for path-dependent options.

Numerical results show improved pricing accuracy in high-dimensional models.

Abstract

We study the problem of reducing the variance of Monte Carlo estimators through performing suitable changes of the sampling measure which are induced by feedforward neural networks. To this end, building on the concept of vector stochastic integration, we characterize the Cameron-Martin spaces of a large class of Gaussian measures which are induced by vector-valued continuous local martingales with deterministic covariation. We prove that feedforward neural networks enjoy, up to an isometry, the universal approximation property in these topological spaces. We then prove that sampling measures which are generated by feedforward neural networks can approximate the optimal sampling measure arbitrarily well. We conclude with a comprehensive numerical study pricing path-dependent European options for asset price models that incorporate factors such as changing business activity, knock-out barriers, dynamic correlations, and high-dimensional baskets.