Mean Field Game of Controls and An Application To Trade Crowding

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.

• The paper introduces an extended Mean Field Game framework focusing on controls to capture collective trading impacts.
• It derives closed-form solutions for optimal liquidation that account for heterogeneous risk aversions and persistent market effects.
• Employing Nash equilibrium and adaptive learning, the study reveals practical insights for market microstructure and regulatory strategies.

Mean Field Game of Controls and An Application To Trade Crowding

The paper explores a novel approach to the optimal liquidation problem through the application of the Mean Field Game (MFG) theory, diverging from traditional models by considering the collective behavior of market participants. In standard models, a large trader interacts with a market characterized by background noise. However, this research introduces a model where traders not only tackle price changes due to fair price fluctuations but also those instigated by other similar market players, thus extending the classical notion of optimal trading to what the authors term as "extended Mean Field Game."

Key Insights and Findings

1. Mean Field Game Framework: The authors propose an "extended MFG" framework where the focus is not solely on the agent's positions, as in traditional models, but significantly on their controls. This change is necessitated by the dynamics of the optimal trading problem where the market price is impacted permanently by collective trading actions.
2. Closed Form Solution: The paper presents a closed-form solution to the formulated problem, providing concrete formulas for traders' strategies under the MFG setup. The solution considers the heterogeneity of participants through varied risk aversions and posits conditions under which traders can operate without immediate knowledge of the entire market state, adapting based on observed behaviors.
3. Nash Equilibrium and Learning: The concept of Nash Equilibrium in the MFG context is crucial, suggesting that equilibria can be realized practically if the game is played repeatedly and participants learn over time. This learning aspect allows market players to adjust their strategies based on the aggregate actions of others, leading to a practical equilibrium without explicit coordination.
4. Theoretical Advances: The paper contributes to theoretical expansions in MFG with first-order smoothing functions and extends results applicable to complex stochastic control problems in finance. It addresses the scenario where agents influence not just immediate prices but create persistent market impacts.
5. Stylized Facts: The study uncovers several noteworthy qualitative outcomes about traders' behaviors and optimal strategies. For instance, it highlights how strong connections among traders (determined by permanent market impact coefficients) modify trading speeds, potentially guiding them towards crowd-following strategies in certain instances.

Practical and Theoretical Implications

• Market Microstructure: The findings strengthen understanding of market microstructure by elucidating how coordinated actions among numerous traders influence asset prices over time, providing insights that could aid in designing more robust trading algorithms and understanding market liquidity.
• Regulation Considerations: The paper's model has regulatory implications, suggesting that monitoring the collective actions of market participants could provide more stable market conditions. This is particularly relevant in preventing adverse strategies that might lead to market inefficiencies or systemic risks.

Future Developments

The proposed framework opens avenues for future research in several directions:

• Extending this model to analyze finite participant cases in more detail.
• Exploring deeper learning mechanisms where market participants utilize complex adaptive strategies.
• Further refining the understanding of heterogeneous preferences and their influence on market dynamics.

Overall, the paper provides a comprehensive analysis of optimizing trading strategies through a sophisticated application of Mean Field Game principles, offering both theoretical insights and practical implications for the field of quantitative finance and beyond.