Discussion Papers 2023

"Mean-field Equilibrium Price Formation with Exponential Utility"

Masaaki Fujii and Masashi Sekine

Revised in January 2025, Forthcoming in Stochastics and Dynamics

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.

Keywords: mean-field game theory, equilibrium price formation, exponential utility, optimal martingale principle, McKean-Vlasov type