Optimal consumption and investment under transaction costs

TL;DR

This paper provides a complete solution to the Merton problem with transaction costs, analyzing how the optimal investment strategy depends on market parameters and costs through a free-boundary differential equation.

Contribution

It offers a novel analytical solution to the Merton problem with transaction costs, characterizing the impact of costs on optimal trading boundaries.

Findings

Solution depends on a quadratic function of parameters

Large transaction costs can make sale boundaries independent of purchase costs

The form of the solution determines well-posedness and the no-transaction region location

Abstract

In this article we consider the Merton problem in a market with a single risky asset and transaction costs. We give a complete solution of the problem up to the solution of a free-boundary problem for a first-order differential equation, and find that the form of the solution (whether the problem is well-posed, whether the problem is well-posed only for large transaction costs, whether the no-transaction wedge lies in the first, second or fourth quadrants) depends only on a quadratic whose co-efficients are functions of the parameters of the problem, and then only through the value and slope of this quadratic at zero, one and the turning point. We find that for some parameter values and for large transaction costs the location of the boundary at which sales of the risky asset occur is independent of the transaction cost on purchases. We give both a mathematical and financial reason for this phenomena.