Optimal Execution under Liquidity Uncertainty

ArXiv ID: 2506.11813 “View on arXiv”

Authors: Etienne Chevalier, Yadh Hafsi, Vathana Ly Vath, Sergio Pulido

Abstract

We study an optimal execution strategy for purchasing a large block of shares over a fixed time horizon. The execution problem is subject to a general price impact that gradually dissipates due to market resilience. This resilience is modeled through a potentially arbitrary limit-order book shape. To account for liquidity dynamics, we introduce a stochastic volume effect governing the recovery of the deviation process, which represents the difference between the impacted and unaffected price. Additionally, we incorporate stochastic liquidity variations through a regime-switching Markov chain to capture abrupt shifts in market conditions. We study this singular control problem, where the trader optimally determines the timing and rate of purchases to minimize execution costs. The associated value function to this optimization problem is shown to satisfy a system of variational Hamilton-Jacobi-Bellman inequalities. Moreover, we establish that it is the unique viscosity solution to this HJB system and study the analytical properties of the free boundary separating the execution and continuation regions. To illustrate our results, we present numerical examples under different limit-order book configurations, highlighting the interplay between price impact, resilience dynamics, and stochastic liquidity regimes in shaping the optimal execution strategy.

Keywords: Optimal Execution, Market Resilience, Price Impact, Stochastic Control, Hamilton-Jacobi-Bellman (HJB) Equations, Equities

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 3.5/10
  • Quadrant: Lab Rats
  • Why: The paper employs advanced mathematical machinery, including singular stochastic control, Hamilton-Jacobi-Bellman variational inequalities, viscosity solutions, and regime-switching Markov chains, which is highly theoretical and dense. While it presents numerical examples, the methodology is primarily analytical, lacking the backtest-ready empirical validation, datasets, or statistical metrics typically found in high-rigor finance papers.
  flowchart TD
    A["Research Goal<br>Determine optimal execution strategy under liquidity uncertainty"] --> B["Methodology<br>Stochastic Control & Singular Control"]
    B --> C["Data/Inputs<br>Limit-Order Book Shapes & Liquidity Regimes"]
    C --> D["Computational Process<br>Solve HJB Variational Inequalities & Analyze Free Boundaries"]
    D --> E["Key Findings/Outcomes<br>Unique Viscosity Solution & Optimal Execution Timing"]