Jörn Saß, Universität Kiel
Up to now numerous technically very elaborate papers dealing with portfolio optimization under transaction costs were published in which transaction costs are defined in three different ways: Proportionally to volume of trade (proportional costs), proportionally to portfolio value (fixed costs) or consisting of a constant component and proportional costs (constant plus proportional costs).
An elegant approach is provided by Morton and Pliska . They show that for the objective of maximizing the expected asymptotic growth rate a factorization of the wealth process is possible that leads under logarithmic utility to an additive representation. Using a reduction to one trading period they only have to solve an optimal stopping problem with linear costs for the portfolio process (fraction of wealth in the stock). It is not obvious if such an approach is possible for more complex and realistic transaction costs.
By reformulating the control problem we prove in the Black-Scholes model that a factorization of the wealth process is indeed possible for general transaction costs. In case of fixed and proportional costs we can reduce the problem in a suitable class of trading strategies to one period between two trading times. This result is proved by renewal theoretic arguments. Finally we obtain an explicit function which has only to be maximized in four paramters to get an optimal strategy in this class.
Our results extend for one stock the results achieved by Morton and Pliska  for fixed costs to combined fixed and proportional costs. The main difference is that we have to distinguish the new proportion of wealth after selling from the proportion of wealth after buying and hence the maximization has to be carried out over an initial distribution.
|||A.J. Morton and S.R. Pliska, 1995: Optimal portfolio management with fixed transaction costs, Mathematical Finance 5, 337-356.|