Unbiased deep solvers for linear parametric PDEs

TL;DR

This paper introduces deep learning algorithms that approximate solutions and gradients of parametric PDEs, enabling bias removal in derivative pricing through Monte Carlo methods, and demonstrating effectiveness in high-dimensional problems.

Contribution

The paper presents novel neural network algorithms that jointly approximate PDE solutions and their gradients, integrating Monte Carlo simulations to eliminate bias, especially useful in high-dimensional settings.

Findings

Algorithms work for high-dimensional problems (up to 100 dimensions)

Method is robust and can be used as a black-box with limited prior info

Provides diagnostics for neural network architecture selection

Abstract

We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the derivative prices and hedging strategies are computed simulatenously. Having approximated the gradient of the solution one can combine it with a Monte-Carlo simulation to remove the bias in the deep network approximation of the PDE solution (derivative price). This is achieved by leveraging the Martingale Representation Theorem and combining the Monte Carlo simulation with the neural network. The resulting algorithm is robust with respect to quality of the neural network approximation and consequently can be used as a black-box in case only limited a priori information about the underlying problem is available. We believe this is important as neural network based algorithms often require fair amount of tuning to produce satisfactory results. The methods are empirically shown to work for high-dimensional problems (e.g. 100 dimensions). We provide diagnostics that shed light on appropriate network architectures.