Mean Field Game of Controls and An Application To Trade Crowding

TL;DR

This paper formulates a mean field game model for optimal trading that accounts for strategic interactions among multiple traders, providing explicit solutions and conditions for learning in complex market environments.

Contribution

It introduces an extended MFG framework for control problems in trading, deriving a closed-form solution and analyzing heterogenous preferences and learning conditions.

Findings

Derived a closed-form solution for the extended MFG of controls.

Analyzed the impact of heterogenous risk preferences on trading strategies.

Established conditions for decentralized learning among traders.

Abstract

In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a "background noise" (or "mean field"). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of "extended MFG", we hence provide generic results to address these "MFG of controls", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of "heterogenous preferences" (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can "learn" it day after day, observing others' behaviors.