Paper: arXiv 2304.07108

Authors: Masaaki Fujii, Masashi Sekine

Abstract

In this paper, using the mean-field game theory, we study a problem of equilibrium price formation among many investors with exponential utility in the presence of liabilities unspanned by the security prices. The investors are heterogeneous in their initial wealth, risk-averseness parameter, as well as stochastic liability at the terminal time. We characterize the equilibrium risk-premium process of the risky stocks in terms of the solution to a novel mean-field backward stochastic differential equation (BSDE), whose driver has quadratic growth both in the stochastic integrands and in their conditional expectations. We prove the existence of a solution to the mean-field BSDE under several conditions and show that the resultant risk-premium process actually clears the market in the large population limit.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly sophisticated theoretical model utilizing mean-field game theory and novel quadratic mean-field BSDEs to establish price formation under exponential utility. However, it is purely theoretical with no empirical backtesting, market data validation, or algorithmic implementations, resulting in very low empirical rigor and reproducibility.

Research Flowchart

  flowchart TD
    A[Goal: Equilibrium Price Formation with Exponential Utility & Heterogeneous Liabilities] --> B[Inputs: Initial Wealth, Risk-Averseness & Unspanned Stochastic Liabilities]
    B --> C[Methodology: Mean-Field Game Framework with Heterogeneous Investors]
    C --> D[Model Formulation: Derivation of novel Mean-Field BSDE with Quadratic Growth Driver]
    D --> E[Mathematical Proof: Existence of Solution for Mean-Field BSDE]
    E --> F[Outcome: Unique Equilibrium Risk-Premium Process Clearing Market in Large Population Limit]