Dynamic consumption and asset allocation with derivative securities

Abstract

This paper provides a model in which a long-term investor chooses consumption and optimal dynamic asset allocation, including a riskless bond, risky stock, and derivatives on the stock, when there is predictable variation in return volatility. We maximize a more general recursive utility function defined over intermediate consumption rather than terminal wealth in order to reflect the realistic problem facing an investor who is saving for the future. The paper obtains a solution to this problem which is exact for investors with unit elasticity of intertemporal substitution of consumption, and approximate otherwise. The optimal log consumption–wealth ratio is a function of stochastic volatility, and the consumption–wealth ratio is an increasing function of volatility for investors whose elasticity of intertemporal substitution of consumption is smaller than one, while it is a decreasing function of volatility for investors whose elasticity of intertemporal substitution of consumption is larger than one. Derivative securities are a significant tool for expanding investors’ dimension of risk-and-return tradeoffs, being a vehicle for the additional risk factor of stochastic volatility in the stock market. Non-myopic investors can utilize derivative securities, which provide access to volatility risk, to capitalize upon the time-varying nature of their opportunity set.

Keywords:

• Intertemporal consumption
• Dynamic asset allocation
• Stochastic opportunities

Notes

†From equation (9) we have , and we also conjecture that there exists a solution of the functional form , and we make the transformation . We thus have and . Therefore, we have