Pure self-financing trading strategies under transaction costs

摘要

We consider a multi-asset discrete-time model of a financial market with proportional transaction costs as described by Schachermayer [7]. In this model, the set of all self-financing trading strategies contains for example trading strategies which consist of buying an asset and selling it at the same time. In the presence of transaction costs this only generates a net loss of investor's money. We introduce the notion of naive pure self-financing trading strategy. Agents using a naive pure self-financing trading strategy do not waste money the way described above. It is well known that the only strategies which do not generate a net loss of investor's money due to transaction costs are the boundary points of the cone of portfolios available at price zero. We call these strategies pure self-financing trading strategies or just pure self-financing strategies. Next, we give a characterization of the set of pure self-financing strategies and of the set of naive pure self-financing strategies. Furthermore, we analyze the relationship between them. Finally, we establish a hedging theorem concerning these strategies.