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

TL;DR

This paper models a market entry and liquidation game with trading constraints, deriving equilibrium strategies through a complex integral equation, and proves the existence and uniqueness of these solutions in both finite and mean-field settings.

Contribution

It introduces a novel mean-field game framework for portfolio liquidation with directional trading constraints and characterizes equilibrium strategies via a complex integral equation.

Findings

Existence of a unique solution to the integral equation

Equilibrium strategies characterized by timing of market entry and exit

Equivalence between N-player and mean-field game formulations

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.