Complete Market Equilibrium with Heterogeneous Agents
We study the market price of risk, the stock volatility and the hedging behavior in equilibrium of heterogeneous agents with arbitrary utility functions, consuming only at the end of the time horizon, and with the state variable following an arbitrary homogeneous diffusion process. We introduce a new notion that we call the ”rate of macroeconomic fluctuations”, and show that, in equilibrium, all the quantities and strategies can be characterized in terms of the dividend volatility and the interest rate volatility discounted at this rate. We also show that both the optimal portfolio strategies and the stock price volatility can be decomposed into a myopic and a non-myopic component. The market price of risk, the myopic volatility and the myopic portfolio are determined by the present market value of future discounted volatilities of the dividend and of the interest rate. By contrast, the non-myopic volatility and non-myopic portfolio are given in terms of covariances of equilibrium quantities with the discounted dividend volatility. These representations enable us to show that, under natural cyclicality conditions, the non-myopic volatility is always positive, and the non-myopic portfolio is positive for an agent if and only if the product of his prudence and risk tolerance is less than the same product corresponding to the log agent.