Relative entropy-regularized robust optimal order execution

ArXiv ID: 2311.06476 “View on arXiv”

Authors: Unknown

Abstract

The problem of order execution is cast as a relative entropy-regularized robust optimal control problem in this article. The order execution agent’s goal is to maximize an objective functional associated with his profit-and-loss of trading and simultaneously minimize the execution risk and the market’s liquidity and uncertainty. We model the market’s liquidity and uncertainty by the principle of least relative entropy associated with the market volume. The problem of order execution is made into a relative entropy-regularized stochastic differential game. Standard argument of dynamic programming yields that the value function of the differential game satisfies a relative entropy-regularized Hamilton-Jacobi-Isaacs (rHJI) equation. Under the assumptions of linear-quadratic model with Gaussian prior, the rHJI equation reduces to a system of Riccati and linear differential equations. Further imposing constancy of the corresponding coefficients, the system of differential equations can be solved in closed form, resulting in analytical expressions for optimal strategy and trajectory as well as the posterior distribution of market volume. Numerical examples illustrating the optimal strategies and the comparisons with conventional trading strategies are conducted.

Keywords: optimal execution, relative entropy, stochastic differential game, Hamilton-Jacobi-Isaacs equation, liquidity modeling, Equities (Trading)

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, featuring advanced stochastic calculus, a relative entropy-regularized Hamilton-Jacobi-Isaacs (rHJI) equation, and analytical solutions derived under linear-quadratic and Gaussian assumptions, resulting in high mathematical complexity. Empirical rigor is low as it only mentions numerical examples and comparisons with conventional strategies without presenting backtests, performance metrics, or implementation details.

  flowchart TD
    A["Research Goal<br>Formulate robust optimal execution<br>using relative entropy regularization"] --> B["Methodology<br>Cast problem as stochastic differential game<br>with relative entropy penalty"]
    B --> C["Key Theoretical Derivation<br>Apply dynamic programming<br>Derive rHJI equation"]
    C --> D["Assumptions & Simplification<br>Linear-Quadratic model<br>Gaussian prior on market volume"]
    D --> E["Computational Process<br>Reduce rHJI to Riccati & linear DEs<br>Assume constant coefficients"]
    E --> F["Closed-Form Solution<br>Derive analytical expressions for:<br>Optimal strategy<br>Optimal trajectory<br>Posterior market volume"]
    F --> G["Key Outcomes<br>Optimal execution strategy found<br>Comparison with conventional strategies<br>Illustrated via numerical examples"]