Portfolio investment with proportional transactions costs and uncertain time horizon: a case of two assets

In this work analysis of optimal control in a portfolio with a deterministic time horizon, uncertain time horizon, and proportional transaction costs are studied. Model equations for deterministic time horizon, non-deterministic time horizon, and non-deterministic horizon with proportional transaction costs are formulated using the technique of Dynamic Programming Principle(DPP), Hamilton JacobiBellman equation, and martingale method. These model equations are solved using normal analytical methods for Ordinary Differential Equation(ODE), Martingale approach to portfolio optimization, and numerical approximation method. Solution obtained by solving the three model equations show that; for the case of deterministic and no-deterministic (with no transaction cost) time horizon the optimal control is constant and independent of the time horizon and initial wealth. While for the case of non-deterministic horizon the optimal control is not constant and their change are determined by two critical numbers that are estimated using numerical methods. These results means that, the fraction of wealth invested in stock is unaffected by the uncertainty in exit time in case there are no transaction costs, and transaction costs reduces the trading frequency for a continuous portfolio investment in order to avoid trading cost from exceeding the value of portfolio returns. This study can be extended to include the trading frequency that can be allowed for a given time in order to keep the optimal wealth. It can also be extended to find out utility functions and variable transformations that may give exact solution to portfolio with transaction costs.