A Deep Neural Network Algorithm for Linear-Quadratic Portfolio Optimization with MGARCH and Small Transaction Costs

TL;DR

This paper develops a neural network-based reinforcement learning algorithm for portfolio optimization under MGARCH models with small transaction costs, providing theoretical error bounds and an efficient expansion method.

Contribution

It introduces a novel neural network fixed-point algorithm with error bounds and an expansion approach for small transaction costs in portfolio optimization.

Findings

The neural network approximation error can be bounded and reduced by increasing parameters.

The expansion method provides a stable, fast, and effective solution for small transaction costs.

The algorithm demonstrates positive testing performance in numerical experiments.

Abstract

We analyze a fixed-point algorithm for reinforcement learning (RL) of optimal portfolio mean-variance preferences in the setting of multivariate generalized autoregressive conditional-heteroskedasticity (MGARCH) with a small penalty on trading. A numerical solution is obtained using a neural network (NN) architecture within a recursive RL loop. A fixed-point theorem proves that NN approximation error has a big-oh bound that we can reduce by increasing the number of NN parameters. The functional form of the trading penalty has a parameter $\epsilon>0$ that controls the magnitude of transaction costs. When $\epsilon$ is small, we can implement an NN algorithm based on the expansion of the solution in powers of $\epsilon$. This expansion has a base term equal to a myopic solution with an explicit form, and a first-order correction term that we compute in the RL loop. Our expansion-based algorithm is stable, allows for fast computation, and outputs a solution that shows positive testing performance.