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 modeling of foreign exchange markets under transaction costs. The financial market is described by a d×d matrix‐valued stochastic process (Πt)Tt=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 (Πt)Tt=0. The main theorem states that the bid‐ask process (Πt)Tt=0 satisfies the robust no‐arbitrage condition iff it admits a strictly consistent pricing system. This result extends the theorems of Harrison‐Pliska and Kabanov‐Stricker pertaining to the case of finite Ω, as well as the theorem of Dalang, Morton, and Willinger and Kabanov, Rásonyi, and Stricker, pertaining to the case of general Ω. An example of a 5 × 5‐dimensional process (Πt)2t=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, Rásonyi, and Stricker.