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

Paper: arXiv 2301.10869

Authors: Andrew Papanicolaou, Hao Fu, Prashanth Krishnamurthy, Farshad Khorrami

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 $ε>0$ that controls the magnitude of transaction costs. When $ε$ is small, we can implement an NN algorithm based on the expansion of the solution in powers of $ε$. 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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous theoretical foundation combining fixed-point theorems, asymptotic expansion, and neural network bounds for dynamic portfolio optimization under stochastic volatility (MGARCH) and transaction costs. While it demonstrates positive numerical test performance, empirical validation on extensive real-world historical data is somewhat limited compared to its mathematical depth.

Research Flowchart

  flowchart TD
    A["Research Goal: Solve Portfolio Mean-Variance Optimization<br/>(MGARCH Dynamics & Small Transaction Costs)"] --> B["Inputs: Asset Return Data, Risk Preferences &<br/>Transaction Cost Parameter (ε)"]
    B --> C["Methodology: Reinforcement Learning & Asymptotic Expansion<br/>(Myopic Base Term + First-Order Correction)"]
    C --> D["Computational Process: Train Neural Network Architecture<br/>in Recursive Fixed-Point Loop"]
    D --> E["Key Outcomes: Fast, Stable Computation &<br/>Bounded Approximation Error O(NN Parameters)"]
    E --> F["Testing Outcome: Strong Out-of-Sample<br/>Portfolio Performance"]