Dynamic portfolio selection with transaction costs: A non-singular stochastic optimal control approach

The dynamic portfolio selection problem with fixed and/or proportional transaction costs is studied. The portfolio consists of a risk-free asset, and a risky asset whose price dynamics is generated by geometric Brownian motion. The objective is to find the controls (amounts invested in the risky and risk-free assets) that maximize the expected value of the discounted utility of the terminal wealth. In contrast to the existing formulations by singular stochastic optimal control theory, the dynamic optimization problem is formulated as a non-singular stochastic optimal control problem so that optimal trading strategies are obtained explicitly. Dynamic programming leads to a Hamilton-Jacobi-Bellman (HJB) equation for the value function. Various (in)equalities are derived in terms of the value function and its derivatives as optimality conditions. Various quasi-variational inequalities, obtained by other researchers using singular and impulse stochastic optimal control formulations, are derived from the optimality conditions. In the absence of transaction costs, the optimal control problem in the new formulation is solved analytically when the portfolio consists of a risk-free asset, and many risky assets whose price dynamics are governed by correlated geometric Brownian motions. The analytical solutions for the investments in the risky assets coincide with those obtained by Merton and others when the utility functions for the terminal wealth belong to the Hyperbolic Absolute Risk Aversion (HARA) and Constant Absolute Risk Aversion (CARA) class of functions.; In the presence of transaction costs, the portfolio space is divided into three disjoint regions, which can be specified as the buying and selling regions of the risky asset and the no transaction region. In the presence of proportional transaction costs, the problem is characterized by two time-dependent curves (buy-no transaction interface and sell-no transaction interface) on the portfolio space. If there are two portfolios that lie in the buying region so that their net values (in terms of the risky asset) remain the same, then the risky asset is bought such that the two rebalancings result in the same portfolio (having the same net value) that lies on the buy-no transaction interface. (Abstract shortened by UMI.)...