Incomplete Continuous-time Securities Markets with Stochastic Income Volatility

TL;DR

This paper derives closed-form solutions for equilibrium interest rates and risk premia in an incomplete continuous-time securities market with stochastic income volatility, highlighting how unspanned income volatility affects market outcomes.

Contribution

It provides the first explicit solutions for equilibrium in incomplete markets with stochastic income volatility and analyzes their impact on interest rates and risk premia.

Findings

Unspanned income with stochastic volatility can lower interest rates.

Such volatility can increase risk premia.

Equilibrium exists under these conditions.

Abstract

In an incomplete continuous-time securities market with uncertainty generated by Brownian motions, we derive closed-form solutions for the equilibrium interest rate and market price of risk 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 countercyclical volatility, the resulting equilibrium can display both lower interest rates and higher risk premia compared to the Pareto efficient equilibrium in an otherwise identical complete market.