Portfoliooptimierung unter Transaktionskosten
Up to now numerous papers dealing with 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). All approaches lead to plausible optimal strategies in view of the used transaction costs. But the results are not satisfieing in so far as either the transaction cost structure seems to be unrealistic or the optimal strategies are not practicable and hardly determinable. Our goal is to derive practicable optimal strategies for logarithmic utility. First, we analyze the CRR model in one period. We introduce a general cost structure containing the transaction cost models described above. Because of the simple structure, this model can be analyzed in a fairly complete way. In the CRR model with an infinite time-horizon we are able to embed our model in the theory of Markov controlled processes in discrete time. We solve some classical optimization problems without transaction costs and prove several existence results for the case that transaction costs are considered. Finally, we turn to the objective of maximizing the asymptotic growth rate in the continuous-time Black-Scholes model under fixed and proportional costs. We are able to prove a factorization of the wealth process. Using this and some renewal theory arguments we can reduce the problem in a suitable class of control strategies to a period between two trading times. If an optimal strategy exists in this class, it depends only on four parameters and can be determined in a numerically easy way. This result extends the results achieved by Morton and Pliska (1995) 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.