Portfolio Selection with Transaction Costs

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.

[1]  A. J. Taylor,et al.  Optimal control of a Brownian storage system , 1978 .

[2]  Alain Bensoussan,et al.  Impulse Control and Quasi-Variational Inequalities , 1984 .

[3]  Michael J. Klass,et al.  A Diffusion Model for Optimal Portfolio Selection in the Presence of Brokerage Fees , 1988, Math. Oper. Res..

[4]  S. Shreve,et al.  Connections between Optimal Stopping and Singular Stochastic Control I. Monotone Follower Problems , 1984 .

[5]  R. C. Merton,et al.  Optimum Consumption and Portfolio Rules in a Continuous-Time Model* , 1975 .

[6]  C. Doléans-Dade,et al.  On the existence and unicity of solutions of stochastic integral equations , 1976 .

[7]  S. Shreve,et al.  Equivalent models for finite-fuel stochastic control , 1986 .

[8]  Suresh P. Sethi,et al.  Explicit Solution of a General Consumption/Investment Problem , 1986, Math. Oper. Res..

[9]  Ruth J. Williams.  Brownian motion in a wedge with oblique reflection at the boundary , 1985 .

[10]  S. Shreve,et al.  Optimal portfolio and consumption decisions for a “small investor” on a finite horizon , 1987 .

[11]  M. Magill,et al.  The preferability of investment through a mutual fund , 1976 .

[12]  Darrell Duffie,et al.  Transactions costs and portfolio choice in a discrete-continuous-time setting , 1990 .

[13]  S. Varadhan,et al.  Diffusion processes with boundary conditions , 1971 .

[14]  G. Constantinides.  Capital Market Equilibrium with Transaction Costs , 1986, Journal of Political Economy.

[15]  Robert F. Anderson,et al.  Small Random perturbation of dynamical systems with reflecting boundary , 1976, Nagoya Mathematical Journal.

[16]  J. Harrison,et al.  Brownian motion and stochastic flow systems , 1986 .

[17]  H. Leland..  Option Pricing and Replication with Transactions Costs , 1985 .

[18]  H. Witsenhausen,et al.  Some solvable stochastic control problemst , 1980 .

[19]  K. Soner.  Portfolio selection with transaction costs , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.