  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.
