Efficient Machine Learning Methods for Solving Hamilton-Jacobi-Bellman Equations in Finance

Abstract

Recent developments in machine learning have allowed for the solution of high-dimensional partial differential equations overcoming the curse of dimensionality. In our work, we utilize deep neural networks to our advantage to solve Hamilton-Jacobi-Bellman (HJB) equations that arise in financial applications efficiently. Two prevalent methods are commonly used to solve high-dimensional HJB equations. One method uses physics-based neural networks to approximate the unknown solution while regularizing the solution with boundary conditions. Another method, and the focus of this work, represents the HJB equation as a backward-stochastic differential equation (BSDE). This allows us to solve the HJB equation using a reinforcement learning framework.

A major bottleneck is present in both methods. Both methods require a neural network to be trained at each timestep. This means that a new network with large number of parameters need to be stored and trained at each timestep which requires a lot of memory, time and energy. In this work we propose different methods to solve the HJB equation in specific financial applications. To do this, we design specific neural network architectures that can fit the problem and derive loss functions to allow us to train efficient networks. In this work we explore four different problems. Firstly, we solve the HJB equation that arises in high-dimensional American options pricing. We present a method that uses two separate neural network that learns to approximate the price and delta. We improve efficiency by reducing the number of neural networks that we need to train. Secondly, we extend and solve the optimal trade execution problem for multiple agents and assets under the Markowitz criteria in a time-consistent way.

Lastly, we solve the inverse problem with minimal arbitrage violations. The computation of implied volatilities from data is a challenging task that is necessary to accurately compute the price and hedging positions of over-the-counter (OTC) products.