The fundamental decision faced by an investor is how to invest among various assets over time. This problem is known as dynamic portfolio selection. Investment decisions share two important characteristics in varying degrees: uncertainty over the future rewards from the investment, and the timing of the investment. In this seminar, the dynamic portfolio selection problem with fixed and/or proportional transaction costs is presented. The portfolio consists of a risk-free asset, and many risky assets whose price dynamics are generated by correlated geometric Brownian motions. The objective is to find the stochastic 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 can be obtained explicitly. In the limiting case of zero transaction costs, the optimal control problem in the new formulation is solved analytically.
The optimal policies are characterized by solving the optimization problem associated with the Hamilton-Jacobi-Bellman equation (corresponding to one risky and one risk-free asset case). In the presence of transaction costs, the portfolio space is divided into buying and selling regions of the risky asset, and the no transaction region. When there are strictly positive fixed and proportional transaction costs, the problem is characterized by four time-dependent curves (sell-no transaction interface, buy-no transaction interface, sell-target and buy-target) in the portfolio space. Numerical results are presented for buy and sell-no transaction interfaces and buy and sell-targets that characterize the optimal policies of a Constant Relative Risk Aversion investor. Some important properties of the optimal policies are as follows: Proportional transaction costs widen the region of no transaction in a skewed way. Merton's (no transaction) line that corresponds to no transaction costs case, need not lie in the no transaction region. As the interest rate dips, the investment in the risky asset increases and the width of no transaction region decreases.