Incomplete Continuous-Time Securities Markets with Stochastic Income Volatility

In an incomplete continuous-time securities market governed by Brownian motions, we derive closed-form solutions for the equilibrium risk-free rate and equity premium processes. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income and can trade continuously on a finite time interval in a money market account and a single risky security. Besides establishing the existence of an equilibrium, our main result shows that if the investors' unspanned income has stochastic counter-cyclical volatility, the resulting equilibrium can display both lower risk-free rates and higher risk premia relative to the Pareto efficient equilibrium in an otherwise identical complete market. Consequently, our model can simultaneously help explaining the risk-free rate and equity premium puzzles.