Optimal consumption and investment with liquid and illiquid assets

TL;DR

This paper analyzes an optimal consumption and investment problem involving one bond, one liquid risky asset, and one illiquid risky asset with transaction costs, providing explicit strategies and conditions for well-posedness.

Contribution

It fully characterizes optimal strategies using a free boundary ODE and identifies conditions for problem well-posedness involving shadow prices.

Findings

Optimal strategies derived from free boundary ODE solutions

Explicit conditions for model well-posedness

Impact of liquid asset trading on investor strategies

Abstract

We consider an optimal consumption/investment problem to maximize expected utility from consumption. In this market model, the investor is allowed to choose a portfolio which consists of one bond, one liquid risky asset (no transaction costs) and one illiquid risky asset (proportional transaction costs). We fully characterize the optimal consumption and trading strategies in terms of the solution of the free boundary ODE with an integral constraint. We find an explicit characterization of model parameters for the well-posedness of the problem, and show that the problem is well-posed if and only if there exists a shadow price process. Finally, we describe how the investor's optimal strategy is affected by the additional opportunity of trading the liquid risky asset, compared to the simpler model with one bond and one illiquid risky asset.