Comparisons and asymptotics in the theory of portfolio optimization under fixed and proportional transaction costs
In this thesis we consider two portfolio optimization models under fixed and proportional transaction costs and investigate their asymptotic behavior. In the Black-Scholes framework of one bond and d ≥ 1 stocks the problem treated is to maximize the expected asymptotic growth rate with logarithmic utility of wealth, when transaction costs, consisting of a fixed part (proportional to the current portfolio value) and a proportional part (proportional to the trading volume) have to be paid. The first model, where the transaction costs are paid from the bond, was introduced and treated by Irle and Sass 2005 and 2006. An optimal strategy for the one-dimensional case was found in the class of constant boundary strategies given by four boundary constants a < α ≤ β < b ∈ ]0,1[, where ]a,b[ is the continuation region for the risky fraction process and α, β denote the states it is traded back to at each transaction. For the second model, where the fixed part of the costs is paid proportionally from each asset, the existence of a general optimal strategy was shown by Tamura 2006 and 2008 via a weak solution to a quasi-variational inequality. By establishing a maximum principle for this solution and additional information on the corresponding continuation region in the general case d ≥ 1, we first obtain the existence of an optimal constant boundary strategy for the second model in the one-dimensional case. The difference of the two models in terms of optimal boundaries and optimal asymptotic growth rates is then analyzed numerically. In the second part we investigate in the one-dimensional case the asymptotic behavior of these two models when the proportional part of the transaction costs tends to zero. The convergence to the model with only fixed costs, introduced by Morton and Pliska 1995, is established with regard to the optimal boundaries and asymptotic growth rates. Moreover the uniqueness of the optimal boundaries for the limit model is obtained. Finally, the convergence of the models, when the fixed costs tend to zero, is treated. A suitable model with purely proportional costs similar to that of Taksar, Klaas, and Assaf 1988 is introduced and an optimal strategy is shown to consist of keeping the risky fraction process in a unique interval [A, B] ⊆ ]0, 1[ with minimal effort. Furthermore, it is proved to be the limit model in terms of optimal boundaries, asymptotic growth rates and optimal risky fraction processes.