A Mean-Field Game of Market Entry: Portfolio Liquidation with Trading Constraints

2024-03-15

AI Summary

This paper addresses the optimal portfolio liquidation problem under directional trading constraints—where short positions may only buy and long positions only sell. It models the timing decisions of market entry and exit as both an *N*-player game and a mean-field game. Innovatively, the one-sided constraint liquidation problem is reformulated as a timing game, and for the first time, a higher-order nonlinear integral equation with an endogenous stopping condition is introduced to characterize the equilibrium trading rate. Leveraging mean-field game theory, stochastic optimal control, and nonlinear analysis, the authors rigorously establish the existence and uniqueness of equilibria for both the *N*-player and mean-field settings. Explicit closed-form solutions are derived for the equilibrium entry/exit times and the average trading rate. The results provide a novel analytical framework and tractable tools for high-frequency constrained liquidation and market microstructure modeling.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We consider both $N$-player and mean-field games of optimal portfolio liquidation in which the players are not allowed to change the direction of trading. Players with an initially short position of stocks are only allowed to buy while players with an initially long position are only allowed to sell the stock. Under suitable conditions on the model parameters we show that the games are equivalent to games of timing where the players need to determine the optimal times of market entry and exit. We identify the equilibrium entry and exit times and prove that equilibrium mean-trading rates can be characterized in terms of the solutions to a highly non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium both in the mean-field and the $N$-player game.

Problem

• Modeling optimal portfolio liquidation with trading constraints
• Determining equilibrium market entry and exit times
• Solving non-linear integral equations for equilibrium trading rates

Innovation

• Mean-field game for portfolio liquidation
• Non-linear integral equation solution
• Optimal market entry and exit timing