The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time

We prove a version of the Fundamental Theorem of Asset Pricing, which applies to Kabanov’s approach to foreign exchange markets under transaction costs. The financial market is modelled by a d × d matrix-valued stochastic process (Σt)t=0 specifying the mutual bid and ask prices between d assets. We introduce the notion of “robust no arbitrage”, which is a version of the no arbitrage concept, robust with respect to small changes of the bid ask spreads of (Σt)t=0. Dually, we interpret a concept used by Kabanov and his co-authors as “strictly consistent price systems”. We show that this concept extends the notion of equivalent martingale measures, playing a well-known role in the frictionless case, to the present setting of bid-ask processes (Σt)t=0. The main theorem states that the bid-ask process (Σt)t=0 satisfies the robust no arbitrage condition iff it admits a strictly consistent pricing system. This result extends the theorems of Harrison-Pliska and Dalang-Morton-Willinger to the present setting, and also generalizes previous results obtained by Kabanov, Rasonyi and Stricker. An example of a 5× 5-dimensional process (Σt)t=0 shows that, in this theorem, the robust no arbitrage condition cannot be replaced by the so-called strict no arbitrage condition, thus answering negatively a question raised by Kabanov, Rasonyi and Stricker.