Mean-Field Price Formation on Trees with Multi-Population and Non-Rational Agents

ArXiv ID: 2510.11261 “View on arXiv”

Authors: Masaaki Fujii

Abstract

This work solves the equilibrium price formation problem for the risky stock by combining mean-field game theory with the binomial tree framework, adapting the classic approach of Cox, Ross & Rubinstein. For agents with exponential and recursive utilities of exponential-type, we prove the existence of a unique mean-field market-clearing equilibrium and derive an explicit analytic formula for equilibrium transition probabilities of the stock price on the binomial lattice. The agents face stochastic terminal liabilities and incremental endowments that depend on unhedgeable common and idiosyncratic factors, in addition to the stock price path. We also incorporate an external order flow. Furthermore, the analytic tractability of the proposed approach allows us to extend the framework in two important directions: First, we incorporate multi-population heterogeneity, allowing agents to differ in functional forms for their liabilities, endowments, and risk coefficients. Second, we relax the rational expectations hypothesis by modeling agents operating under subjective probability measures which induce stochastically biased views on the stock transition probabilities. Our numerical examples illustrate the qualitative effects of these components on the equilibrium price distribution.

Keywords: Mean-Field Game Theory, Binomial Tree Model, Equilibrium Price Formation, Stochastic Optimal Control, Recursive Utility, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced mathematics, including mean-field game theory, FBSDEs, and McKean-Vlasov equations, to derive explicit analytic formulas for equilibrium transition probabilities on a binomial tree. While the theoretical framework is developed for empirical application, the excerpt and summary focus entirely on theoretical derivations and numerical examples for illustration, without any mention of backtesting, real data, or implementation details for live trading.

  flowchart TD
    A["Research Goal:<br/>Equilibrium Price Formation<br/>on Binomial Trees"] --> B["Model Setup:<br/>Mean-Field Game &<br/>Binomial Tree Framework"]
    B --> C["Key Inputs:<br/>Heterogeneous Agents<br/>Stochastic Endowments<br/>External Order Flow"]
    C --> D["Computational Process:<br/>Stochastic Optimal Control<br/>& Market Clearing Condition"]
    D --> E["Outcome 1:<br/>Explicit Analytic Formula<br/>for Equilibrium Probabilities"]
    D --> F["Outcome 2:<br/>Subjective Views<br/>Non-Rational Agents"]
    E --> G["Validation:<br/>Numerical Examples<br/>& Qualitative Analysis"]
    F --> G