Non-Redundant Derivatives in a Dynamic General Equilibrium Economy

Abstract

In this article, we wish to understand: (i) the valuation of a non-redundant derivative in an economy where agents are heterogenous, (ii) the role of such a derivative in an investor's dynamic portfolio strategy, and (iii) the effect of introducing this derivative on the prices of more primitive securities, such as a stock and bond. We do this by studying a dynamic general equilibrium exchange economy in continuous time with two agents who differ in their degree of risk aversion, and where both the endowment and its growth rate are stochastic. We study two versions of this economy: in the first, only a stock and a zero-supply instantaneously riskless bond are available for trading, so that financial markets are incomplete; in the second version of this economy, we introduce a derivative that allows the agent to hedge perfectly the risk arising from the stochastic growth rate of endowment. Our main contribution is to characterize in closed form (using asymptotic analysis) the equilibrium in these two versions of the economy, including an expression for the price of the derivative in the second economy. We then compare analytically the portfolio policies and prices across the two versions of the economy. We find that the introduction of a derivative leads to an increase in the interest rate, expected return on the stock and the volatility of stock returns.