Two methods for optimal investment with trading strategies of finite variation

Two methods for designing optimal portfolios are proposed. In order to reduce the variation in the asset holdings and hence the eventual proportional transaction costs, the trading strategies of these portfolios are constrained to be of a finite variation. The first method minimizes an upper bound on the discrete-time logarithmic error between a reference portfolio and the one with a constrained trading strategy and thus penalizes the shortfall only. A quadratic penalty on the logarithmic variation of the trading strategy is also included in the objective functional. The second method minimizes a sum of the discrete-time log-quadratic errors between the asset holding values of the constrained portfolio and a certain reference portfolio, which results in tracking the reference portfolio. The optimal trading strategy is obtained in an explicit closed form for both methods. Simulation examples with the log-optimal and the Black Scholes replicating portfolios as references show smoother trading strategies for the new portfolios and a significant reduction in the eventual proportional transaction cost. The performance of the new portfolios are very close to their references in both cases.