Optimal position-building strategies in competition

Paper: arXiv 2409.03586

Abstract

This paper develops a mathematical framework for building a position in a stock over a fixed period of time while in competition with one or more other traders doing the same thing. We develop a game-theoretic framework that takes place in the space of trading strategies where action sets are trading strategies and traders try to devise best-response strategies to their adversaries. In this setup trading is guided by a desire to minimize the total cost of trading arising from a mixture of temporary and permanent market impact caused by the aggregate level of trading including the trader and the competition. We describe a notion of equilibrium strategies, show that they exist and provide closed-form solutions.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced mathematics including game theory, differential equations, and the Euler-Lagrange equation, but it focuses on theoretical derivations and closed-form solutions without providing backtesting frameworks or data-driven implementation details.

Research Flowchart

  flowchart TD
  A["Research Goal<br/>Find optimal competitive position-building strategies<br/>minimizing total trading cost"] --> B
  subgraph B ["Methodology"]
      B1["Develop game-theoretic framework<br/>strategies vs. strategies"]
      B2["Model market impact<br/>temporary & permanent"]
      B3["Formulate cost minimization problem"]
  end
  B --> C["Inputs<br/>Fixed time horizon, initial/final inventory,<br/>market impact parameters, competitor behavior"]
  C --> D["Computation<br/>Derive Euler-Lagrange equations<br/>Solve for best-response strategies"]
  D --> E["Outcome<br/>Existence of Nash Equilibrium<br/>Closed-form solution for optimal strategies<br/>Optimal execution rate formula"]

Cited References (16)

1. Optimal execution of portfolio transactions (2001)
2. Optimal liquidation (1997)
3. Calculus of variations (2000)
4. Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization (1999)
5. Market reaction to temporary liquidity crises and the permanent market impact (2006)
6. Direct estimation of equity market impact (2005)
7. No-dynamic-arbitrage and market impact (2010)
8. Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework (2011)

0 of 16 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.