A super-replication theorem in Kabanov’s model of transaction costs

Abstract

We prove a general version of the super-replication theorem, which applies to Kabanov’s model of foreign exchange markets under proportional transaction costs. The market is described by a matrix-valued càdlàg bid-ask process $(\Pi_t)_{t\in [0,T]}$ evolving in continuous time. We propose a new definition of admissible portfolio processes as predictable (not necessarily right- or left- continuous) processes of finite variation related to the bid-ask process by economically meaningful relations. Under the assumption of existence of a strictly consistent price system (SCPS), we prove a closedness property for the set of attainable vector-valued contingent claims. We then obtain the super-replication theorem as a consequence of that property, thus generalizing to possibly discontinuous bid-ask processes analogous results obtained by Kabanov (Financ. Stoch. 3, 237–248, 1999), Kabanov and Last (Math. Financ. 12, 63–70, 2002) and Kabanov and Stricker (Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, pp 125–136, 2002). Rásonyi’s counter-example (Lecture Notes in Mathematics 1832, 394–398, 2003) served as an important motivation for our approach.