Optimal position-building strategies in competition

ArXiv ID: 2409.03586 “View on arXiv”

Authors: Unknown

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.

Keywords: Trading Strategy, Game Theory, Market Impact, Optimal Execution, Equilibrium Strategies

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: 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.

  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"]