Leveraging IS and TC: Optimal order execution subject to reference strategies

ArXiv ID: 2401.03305 “View on arXiv”

Authors: Unknown

Abstract

The paper addresses the problem of meta order execution from a broker-dealer’s point of view in Almgren-Chriss model under execution risk. A broker-dealer agency is authorized to execute an order of trading on some client’s behalf. The strategies that the agent is allowed to deploy is subject to a benchmark, referred to as the reference strategy, regulated by the client. We formulate the broker’s problem as a utility maximization problem in which the broker seeks to maximize his utility of excess profit-and-loss at the execution horizon, of which optimal feedback strategies are obtained in closed form. In the absence of execution risk, the optimal strategies subject to reference strategies are deterministic. We establish an affine structure among the trading trajectories under optimal strategies subject to general reference strategies using implementation shortfall (IS) and target close (TC) orders as basis. Furthermore, an approximation theorem is proposed to show that with small error, general reference strategies can be approximated by piece-wise constant ones, of which the optimal strategy is piece-wise linear combination between IS and TC orders. We conclude the paper with numerical experiments illustrating the trading trajectories as well as histograms of terminal wealth and utility at investment horizon under optimal strategies versus those under TWAP strategies.

Keywords: Almgren-Chriss Model, Meta Order Execution, Implementation Shortfall, Utility Maximization, Optimal Execution, Equities

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper presents a stochastic control formulation with multiple stochastic differential equations, deriving closed-form solutions and establishing an affine structure theorem, which indicates advanced mathematical density. However, it lacks any mention of real-world data, backtests, or implementation code, relying instead on theoretical numerical examples for illustration.

  flowchart TD
    A["Research Goal:<br>Optimal Execution Subject to<br>Reference Strategy"] --> B["Methodology:<br>Broker Utility Maximization<br>(PnL at Horizon)"]
    B --> C["Model Setup:<br>Almgren-Chriss Model<br>with Execution Risk"]
    C --> D["Key Theoretical Result:<br>Affine Structure:<br>Strategy = α × IS + (1-α) × TC"]
    D --> E["Computational Process:<br>Solve Optimization for<br>Coefficient α"]
    E --> F["Findings/Optimal Trajectory:<br>Subject to Reference Strategy<br>(vs. TWAP Benchmark)"]
    F --> G["Outcomes:<br>1. Closed-form Solution<br>2. High Utility/PnL<br>3. Risk-Adjusted Performance"]