Paper: arXiv 2409.15459

Abstract

This paper extends the optimal-trading framework developed in arXiv:2409.03586v1 to compute optimal strategies with real-world constraints. The aim of the current paper, as with the previous, is to study trading in the context of multi-player non-cooperative games. While the former paper relies on methods from the calculus of variations and optimal strategies arise as the solution of partial differential equations, the current paper demonstrates that the entire framework may be re-framed as a quadratic programming problem and cast in this light constraints are readily incorporated into the calculation of optimal strategies. An added benefit is that two-trader equilibria may be calculated as the end-points of a dynamic process of traders forming repeated adjustments to each other’s strategy.

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 is heavily theoretical, featuring advanced mathematics like variational methods, PDEs, and convex quadratic programming, but lacks any mention of code, backtests, or empirical data, focusing purely on computational methods for theoretical solutions.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Optimal Trading Strategies<br>with Real-World Constraints"] --> B["Methodology:<br>Formulate as Quadratic Programming Problem"]
  B --> C["Incorporate Constraints:<br>Order size, Liquidity, Risk"]
  C --> D["Compute Solutions:<br>Numerical Optimization"]
  D --> E["Dynamic Equilibrium:<br>Simulate Repeated Adjustments"]
  E --> F["Outcome:<br>Equilibrium Strategies &<br>Multi-Trader Interactions"]

Cited References (23)

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  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)
and 15 more
  1. Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate (2015)
  2. Optimal trading strategy and supply/demand dynamics (2013)
  3. Three models of market impact and Data, High-Frequency (2010)
  4. Handbook of price impact modeling (2023)
  5. Dynamical models of market impact and algorithms for order execution (2013)
  6. The market impact of a limit order (2012)
  7. Optimal Execution: A Review (2022) — link
  8. Mathematica, V ersion 14.1 — link
  9. Optimal position-building in competition (2024) — link
  10. Theory of games and economic behavior: 60th anniversary commemorative edition (2007)
  11. Equilibrium points in n-person games (1950)
  12. The Nash Equilibrium: A Perspective (2004) — link
  13. Two-person cooperative games (1953)
  14. Equilibrium Points in N-Person Games (1950) — link
  15. Non-Cooperative Games (1951) — link

0 of 23 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.