Portfolio Selection with Transaction Costs

TLDR

Merton and others solved the optimal consumption and investment problem when transactions between bank and stock are costless. This paper studies optimal consumption and investment decisions for an investor with a bank account and a log‑normal stock. We model transaction costs as a fixed percentage of each trade and solve the resulting nonlinear free‑boundary problem, providing an algorithm for its computation. The optimal buying and selling policies are the local times of the bank‑stock holdings process at the boundaries of a wedge‑shaped region defined by the free‑boundary solution.

Abstract

In this paper, optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log-normal diffusion. This problem was solved by Merton and others when transactions between bank and stock are costless. Here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted. It is shown that the optimal buying and selling policies are the local times of the two-dimensional process of bank and stock holdings at the boundaries of a wedge-shaped region which is determined by the solution of a nonlinear free boundary problem. An algorithm for solving the free boundary problem is given.