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Mean Field Equilibrium Asset Pricing Models With Exponential Utility

Masashi Sekine

Abstract

This thesis develops equilibrium asset pricing models in incomplete markets with a large number of heterogeneous agents using mean field game theory. The market equilibrium is characterized by a novel form of mean field backward stochastic differential equations (BSDEs). First, we propose a theoretical model that endogenously derives the equilibrium risk premium. Agents with exponential preferences are heterogeneous in initial wealth, risk aversion, and unspanned stochastic terminal liability. We solve the optimal investment problem using the optimal martingale principle. The equilibrium is characterized by a mean field BSDE whose driver has quadratic growth in both the stochastic integrands and their conditional expectations. We prove the existence of solutions and show that the risk premium clears the market in the large population limit. Second, we extend the model to include consumption and habit formation, relaxing the time-separability assumption of utility functions. A similar mean field BSDE is derived, and its well-posedness and asymptotic behavior are examined. We also introduce an exponential quadratic Gaussian (EQG) reformulation to obtain equilibrium solutions in semi-analytic form. Finally, the model is extended to partially observable markets where agents must infer the risk premium from stock price observations to determine trading strategies. We provide semi-analytic expressions for the equilibrium via the EQG framework, and the equilibrium risk-premium process is constructed endogenously using Kalman-Bucy filtering theory. Numerical simulations are included to visualize the resulting market dynamics.

Mean Field Equilibrium Asset Pricing Models With Exponential Utility

Abstract

This thesis develops equilibrium asset pricing models in incomplete markets with a large number of heterogeneous agents using mean field game theory. The market equilibrium is characterized by a novel form of mean field backward stochastic differential equations (BSDEs). First, we propose a theoretical model that endogenously derives the equilibrium risk premium. Agents with exponential preferences are heterogeneous in initial wealth, risk aversion, and unspanned stochastic terminal liability. We solve the optimal investment problem using the optimal martingale principle. The equilibrium is characterized by a mean field BSDE whose driver has quadratic growth in both the stochastic integrands and their conditional expectations. We prove the existence of solutions and show that the risk premium clears the market in the large population limit. Second, we extend the model to include consumption and habit formation, relaxing the time-separability assumption of utility functions. A similar mean field BSDE is derived, and its well-posedness and asymptotic behavior are examined. We also introduce an exponential quadratic Gaussian (EQG) reformulation to obtain equilibrium solutions in semi-analytic form. Finally, the model is extended to partially observable markets where agents must infer the risk premium from stock price observations to determine trading strategies. We provide semi-analytic expressions for the equilibrium via the EQG framework, and the equilibrium risk-premium process is constructed endogenously using Kalman-Bucy filtering theory. Numerical simulations are included to visualize the resulting market dynamics.
Paper Structure (50 sections, 36 theorems, 530 equations, 5 figures)

This paper contains 50 sections, 36 theorems, 530 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. A Review of the Recent Research
  4. Preliminary
  5. Mean Field Game Approach to Equilibrium Pricing With Market Clearing Condition
  6. Overview
  7. Comparison With This Research
  8. Mean Field Equilibrium Asset Pricing Model in an Incomplete Market
  9. Preliminary
  10. Exponential Utility Optimization for a Given Agent
  11. The Market and the Utility Function
  12. Characterization of the Optimal Trading Strategy
  13. Solution of the BSDE and Its Verification
  14. Mean-Field Equilibrium Model
  15. Heuristic Derivation of the Mean-Field BSDE
  16. ...and 35 more sections

Key Result

Theorem 3.2.3

(Cited from fujiiMeanFieldGame2022 [Theorem 4.2]) Under Assumptions rev-asm-1 and rev-asm-2, there exists a unique strong solution $(X,Y,Z^0,Z)\in \mathbb{S}^2(\mathbb{F}^{0,1},\mathbb{R}^n) \times \mathbb{S}^2(\mathbb{F}^{0,1},\mathbb{R}^n)\times \mathbb{H}^2(\mathbb{F}^{0,1},\mathbb{R}^{n\times d_

Figures (5)

  • Figure 1: Solutions of Eq.\ref{['Riccati eqn']}
  • Figure 2: Market risk-premium process
  • Figure 3: Asymptotic market clearing
  • Figure 4: Initial and terminal wealth
  • Figure 5: Terminal liability and net asset

Theorems & Definitions (85)

  • Theorem 3.2.3
  • Theorem 3.2.4
  • Remark 4.2.1.2
  • Remark 4.2.1.3
  • Definition 4.2.1.4
  • Remark 4.2.1.5
  • Remark 4.2.1.7
  • Definition 4.2.1.8
  • Remark 4.2.1.9
  • Definition 4.2.2.1
  • ...and 75 more