Mean-Field Price Formation on Trees with a Network of Relative Performance Concerns

Abstract

Financial firms and institutional investors are routinely evaluated based on their performance relative to their peers. These relative performance concerns significantly influence risk-taking behavior and market dynamics. While the literature studying Nash equilibrium under such relative performance competitions is extensive, its effect on asset price formation remains largely unexplored. This paper investigates mean-field equilibrium price formation of a single risky stock in a discrete-time market where agents exhibit exponential utility and relative performance concerns. Unlike existing literature that typically treats asset prices as exogenous, we impose a market-clearing condition to determine the price dynamics endogenously within a relative performance equilibrium. Using a binomial tree framework, we establish the existence and uniqueness of the market-clearing mean-field equilibrium in both single- and multi-population settings. Finally, we provide illustrative numerical examples demonstrating the equilibrium price distributions and agents' optimal position sizes.

Summary

This paper investigates the equilibrium price formation of a single risky stock in a discrete-time market where agents exhibit exponential utility and relative performance concerns. The research addresses the gap in existing literature, which largely treats asset prices as exogenous, by imposing a market-clearing condition to endogenously determine price dynamics within a relative performance equilibrium. The authors utilize a binomial tree framework to model the market and establish the existence and uniqueness of the market-clearing mean-field equilibrium (MC-MFE) in both single- and multi-population settings. They derive explicit solutions based on backward induction and present numerical examples to illustrate equilibrium price distributions and agents' optimal position sizes. The key contribution is extending the discrete-time framework to incorporate relative performance concerns, allowing agents to benchmark their performance against specific peer groups rather than the market average. This is achieved by introducing a heterogeneous network of relative performance concerns represented by θⁱ_p,k, denoting the sensitivity of agent-i in population p relative to the performance of population k. The paper demonstrates that the equilibrium solutions remain continuous even around singular points where the interaction matrix (I − Θ) exhibits a first-order pole, resolving a singularity issue observed in previous relative performance game models. This work matters to the field of financial economics as it provides a tractable framework for analyzing the impact of relative performance concerns on asset pricing, which is a prevalent feature of financial markets, especially among institutional investors.

Key Insights

• •Novelty: The paper is the first to combine mean-field game theory with a binomial tree framework to endogenously determine asset prices in a market with relative performance concerns.
• •Market-Clearing Condition: Imposing the market-clearing condition prevents agents from taking infinitely large positions, which is crucial for realistic market modeling.
• •Continuity Around Singularity: The model demonstrates that the equilibrium solutions remain continuous even around the singular point E[θ] = 1 in the single population setting, and near first-order poles in the multi-population setting, where previous models would break down.
• •Explicit Solutions: The authors derive explicit solutions for the optimal trading strategies and equilibrium transition probabilities, making the model tractable for analysis and computation.
• •Multi-Population Framework: The extension to multiple populations with a network of relative performance concerns (θⁱ_p,k) allows for more realistic modeling of financial markets where agents benchmark against specific peer groups.
• •Equilibrium Transition Probabilities: The equilibrium transition probabilities of the stock price (p_n-1(s,y)) are explicitly derived as a function of the aggregate risk tolerance, market supply, and agents' relative performance concerns.

Practical Implications

• •Risk Management: Financial institutions can use this model to better understand how relative performance concerns influence risk-taking behavior and asset pricing, leading to improved risk management strategies.
• •Portfolio Optimization: Asset managers can leverage the derived optimal trading strategies to construct portfolios that account for relative performance benchmarks and market-clearing conditions.
• •Policy Implications: Regulators can use this framework to analyze the potential impact of policies aimed at reducing excessive risk-taking driven by relative performance concerns.
• •Market Microstructure: The model can be extended to study the effects of market microstructure features, such as order flow and liquidity, on equilibrium price formation in the presence of relative performance concerns.
• •Future Research: The framework can be extended to incorporate more complex asset structures, heterogeneous beliefs, and dynamic learning.

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