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

Paper: arXiv 2401.03305

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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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"]

Cited References (34)

1. Stochastic controls: Hamiltonian systems and HJB equations (2012)
2. Linear-exponential-quadratic Gaussian control (2013)
3. Optimal execution of portfolio transactions (2001)
4. Execution costs (2010)
5. Target close execution strategies — link
6. The Financial Mathematics of Market Liquidity: From optimal execution to market making (2016)
7. Optimal execution of a VWAP order: a stochastic control approach (2015)
8. VWAP execution and guaranteed VWAP (2014)

1 of 34 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.