Relative Arbitrage Opportunities in an Extended Mean Field System

2023-11-05

AI Summary

This paper investigates the existence and asymptotic behavior of relative arbitrage opportunities in markets with competitive investors. Method: For an infinite population of interacting investors, we construct a coupled system governed by conditional McKean–Vlasov equations and introduce, for the first time, an extended mean-field game framework wherein the state space incorporates the joint distribution of wealth and investment strategies. Equilibrium in this model is defined as the optimal relative arbitrage strategy. Contribution/Results: We rigorously establish the existence and uniqueness of such a mean-field equilibrium and prove propagation of chaos together with convergence of the finite-player Nash equilibria to the mean-field limit as the number of agents tends to infinity. Our analysis yields sufficient conditions for the existence of relative arbitrage and provides the first mean-field framework for arbitrage modeling in large-scale financial markets that simultaneously ensures theoretical rigor and structural tractability.

Abstract

This paper studies relative arbitrage opportunities in a market with infinitely many interacting investors. We establish a conditional McKean-Vlasov system to study the market dynamics coupled with investors. We then provide a theoretical framework to study a mean-field system, where the mean-field terms consist of a joint distribution of wealth and strategies. The optimal relative arbitrage is characterized by the equilibrium of extended mean-field games. We show the conditions on the existence and the uniqueness of the mean field equilibrium, then prove the propagation of chaos results for the finite-player game, and demonstrate that the Nash equilibrium converges to the mean field equilibrium when the population grows to infinity.

Problem

• Studying relative arbitrage opportunities in competitive markets with infinite players
• Establishing conditional McKean-Vlasov system for market dynamics analysis
• Characterizing optimal arbitrage as solution to Cauchy PDE with volatility

Innovation

• Extended mean-field game models competitive investors
• Conditional McKean-Vlasov system captures market dynamics
• Cauchy PDE characterizes optimal arbitrage opportunities