Optimal Consumption and Portfolio Choice with No-Borrowing Constraint in the Kim-Omberg Model: The Complete Market Case

TL;DR

This paper analyzes optimal consumption and portfolio strategies under a no-borrowing constraint within the Kim-Omberg model, incorporating stochastic mean-reverting expected returns and solving via duality and optimal stopping methods.

Contribution

It introduces a novel dual control approach to handle the no-borrowing constraint in a complete market setting with stochastic mean reversion.

Findings

Derived explicit optimal policies for consumption and portfolio.

Showed the impact of the no-borrowing constraint on investment behavior.

Provided numerical insights into the model's economic implications.

Abstract

In this paper, we study an intertemporal utility maximization problem in which an investor chooses consumption and portfolio strategies in the presence of a stochastic factor and a no-borrowing constraint. In the spirit of the Kim-Omberg model, the stochastic factor represents the expected excess return of the risky asset. It is perfectly negatively correlated with shocks to the risky asset, and follows an Ornstein-Uhlenbeck process, thereby capturing the mean reversion of expected excess returns-a feature well supported by empirical evidence in financial markets. The investor seeks to maximize expected utility from consumption, subject to the constraint that wealth remains nonnegative at all times. To address the dynamic no-borrowing constraint, we use Lagrange duality to transform the primal problem into a singular control problem in the dual space. We then characterize the solution to the dual singular control problem via an auxiliary two-dimensional optimal stopping problem featuring stochastic volatility, and subsequently retrieve the primal value function as well as the optimal portfolio and consumption plans. Finally, a numerical study is conducted to derive economic and financial implications.