Discrete-Time Portfolio Optimization with Transaction Costs

This paper studies a finite-horizon optimal investment problem with proportional and fixed transaction costs. This is a variant of the classical investment-consumption problem formulated by Robert Merton in the absence of transaction costs. It is composed of two parts. In the first part, we examine the optimal decision to sell or buy a stock at discrete-time intervals in the presence of proportional transaction costs only. This is cast as a singular stochastic control problem. For this problem, we study two constant relative risk aversion utility functions of the investor, namely the logarithm and power utility functions. Our proposed numerical solutions indicate that the solution space can be divided into three distinct regions: buy region, sell region, and no transaction region. The buy region is found to vanish towards the end of the maturity. Parameters such as rate of return, maturity, transaction frequency, and risk aversion coefficient are shown to have an effect to a varying degree on the shape and sensitivity of the boundaries. The numerical results for the discrete-time transaction are shown to converge to the continuous-time solution. In the second part, we present numerical solutions for the discrete-time transaction in the presence of both fixed and proportional transaction costs. This is cast as an impulse stochastic control problem. We show that the governing PDE cannot be reduced in dimension. Instead, we show that it is unbounded in two dimensions. Apart from the three regions, there are also two target amount lines between the buy and sell boundaries. When both fixed and proportional costs are present, the investor buys up to a target amount and sells up to another distinct target amount. When there are only fixed costs, the investor buys and sells up to the same target amount, and this amount is shown to be on the so-called Merton line.Further analysis of the results reveals that the presence of transaction costs significantly affects the investors' transaction strategy, including when to transact and the quantity to trade at each point. Based on a continuous transaction case, we formulate part of the governing PDEs for both problems with a reduction in dimension and propose conditions for the jump. We also solve an optimization problem, present and implement the relevant numerical schemes for solving the governing PDE, together with suitable Neumann and Dirichlet boundary conditions. Detailed analysis of the shape and sensitivity of the boundaries to different parameters are given in the paper. Analysis on the exact amount of transaction due to different types and levels of transaction costs are also provided.

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