No arbitrage and closure results for trading cones with transaction costs

摘要

In this paper, we consider trading with proportional transaction costs as in Schachermayer’s paper (Schachermayer in Math. Finance 14:19–48, 2004). We give a necessary and sufficient condition for ${mathcal{A}}$ , the cone of claims attainable from zero endowment, to be closed. Then we show how to define a revised set of trading prices in such a way that, firstly, the corresponding cone of claims attainable for zero endowment, ${tilde{ {mathcal{A}}}}$ , does obey the fundamental theorem of asset pricing and, secondly, if ${tilde{ {mathcal{A}}}}$ is arbitrage-free then it is the closure of ${mathcal{A}}$ . We then conclude by showing how to represent claims.