Paper: arXiv 2311.02690

Authors: Nicole Tianjiao Yang, Tomoyuki Ichiba

Abstract

This paper studies relative arbitrage opportunities in a market with competitive investors through stochastic differential games in the limit as the number of players tends to infinity. With common noises introduced by the stock capitalization processes, we establish a conditional McKean-Vlasov system to study the market dynamics coupled to the expected trading volume of investors. We show that optimal arbitrage can be characterized as a solution of a Cauchy PDE constructed by the volatility terms in the market model. The structure of the market dynamics can be relaxed, and we provide a theoretical framework to study a general mean-field system, where the interaction is characterized by a joint distribution of wealth and strategies. In this setting, the optimal relative arbitrage constitutes the strong equilibrium of an extended mean-field game. We provide conditions for the existence and uniqueness of the mean-field equilibrium. We further prove the propagation of chaos result for the finite-player game counterpart, and demonstrate that the Nash equilibrium converges to the mean field equilibrium when the population grows to infinity.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly sophisticated theoretical framework using extended mean field games, stochastic differential equations, and conditional McKean-Vlasov systems to analyze relative arbitrage. However, it is purely theoretical and lacks empirical backtesting or real-world market data validation.

Research Flowchart

  flowchart TD
    A[Goal: Analyze Relative Arbitrage in Extended Mean-Field Systems] --> B[Inputs: Market Volatility & Capitalization Processes with Common Noise]
    B --> C[Establish Conditional McKean-Vlasov System]
    C --> D[Solve Cauchy PDE for Optimal Arbitrage Strategy]
    D --> E[Extend to Extended MFG: Joint Distribution of Wealth & Strategies]
    E --> F[Prove Existence/Uniqueness of MFE & Propagation of Chaos]
    F --> G[Outcome: Finite-Player Nash Equilibrium Converges to Strong MFE]