Chi-fu Huang

OPTIMAL CONSUMPTION AND PORTFOLIO POLICIES WHEN ASSET PRICES FOLLOW A DIFFUSION PROCESS

Abstract We consider a consumption-portfolio problem in continuous time under uncertainty. A martingale technique is employed to characterize optimal consumption-portfolio policies when there exist nonnegativity constraints on consumption and on final wealth. We also provide a way to compute and verify optimal policies. Our verification theorem for optimal policies involves a linear partial differential equation, unlike the nonlinear partial differential equation of dynamic programming. The relationship between our approach and dynamic programming is discussed. We demonstrate our technique by explicitly computing optimal policies in a series of examples. In particular, we solve the optimal consumption-portfolio problem for hyperbolic absolute risk aversion utility functions when the asset prices follow a geometric Brownian motion. The optimal policies in this case are no longer linear when nonnegativity constraints on consumption and on final wealth are included. By these examples, one can see that our approach is much easier than the dynamic programming approach.

A variational problem arising in financial economics

Abstract We provide sufficient conditions for a dynamic consumption–portfolio problem in continuous time to have a solution. When the price processes satisfy a regularity condition, all utility functions that are continuous, increasing, concave, and are dominated by a strictly concave power function admit a solution.

Multiperiod security markets with differential information : Martingales and resolution times

We model multiperiod securities markets with differential information. A price system that admits no free lunches is related to martingales when agents have rational expectations. We introduce the concept of resolution time, and show that a better informed agent and a less informed agent must agree on the resolution times of commonly marketed events if they have rational expectations and if there are no free lunches. It then follows that if all elementary events are marketed for a less informed agent then any price system that admits no free lunches to a better informed agent must eliminate any private information asymmetry between the two. We provide an example of a dynamically fully revealing price system that is arbitrage free and yields elementarily complete markets.

Intertemporal preferences for uncertain consumption : a continuous time approach

We propose a family of topologies on the space of consumption patterns in continuous time under uncertainty. Preferences continuous in any of the proposed topologies treat consumptions at nearby dates as almost perfect substitutes except possibly at information surprises. The topological duals of the family of proposed topologies essentially contain processes that are the sums of processes of absolutely continuous paths and martingales. Thus if equilibrium prices for consumption come from the duals, consumptions at nearly adjacent dates in a state of nature have almost equal prices except possibly at information surprises. In particular, if the information structure is generated by a Brownian motion, the duals are composed of Ito processes. We investigate some implications of our topologies on standard models of choice in continuous time as well as on recent models of non-time-separable representations of preferences. We also discuss the properties of prices of long-lived assets in economies populated with agents whose preferences are continuous in our topologies when there are no arbitrage opportunities.

On intertemporal preferences in continuous time: The case of certainty☆

Abstract Different topologies on the space of certain consumption patterns in a continuous time setting are discussed. A family of topologies which give an economically reasonable sense of closeness and have an appropriate intertemporal flavor is suggested. The topological duals of our suggested topologies are essentially spaces of Lipschitz continuous functions. Any utility functional whose felicity function at any time t depends explicitly on the consumption at that time and is not linear in it is not continuous in any one of our topologies. A class of utility functionals that are continuous in the suggested topologies and that capture the intuitively appealing notion that consumptions at nearby dates are almost perfect substitutes is provided. We then give necessary and sufficient conditions for a consumption plan to be optimal for this class of utility functionals. We demonstrate our general theory by solving in closed form the optimal consumption problem for a particular utility function. The optimal solution consists of a (possible) initial ‘gulp’ of consumption, or an initial period of no consumption, followed by consumption at the rate that maintains a constant ratio of wealth to an index of past consumption experience.

A continuous-time portfolio turnpike theorem

Abstract We prove a continuous-time portfolio turnpike theorem. The proof uses the theory of martin-gales and is more intuitively appealing than the usual discrete-time mode of proof using dynamic programming. When the interest rate is strictly positive, the present value of any contingent claim having payoffs bounded from above can be made arbitrarily small when the investment horizon increases. Thus an investor concentrates his wealth in buying contingent claims that have payoffs unbounded from above at the very beginning of his horizon. As a consequence, it is the asymptotic property of his utility function as wealth goes to infinity that determines his optimal investment strategy at the very beginning of his horizon.