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

Abstract

We prove a version of the Fundamental Theorem of Asset Pricing, which applies to Kabanov's modelling of foreign exchange markets under transaction costs. The financial market is described by a d × d matrix-valued stochastic process (Π t ) T 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 (Π t ) T t=0 .The main theorem states that the bid-ask process (Π t ) T 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 Kabanov-Stricker pertaining to the case of finite Ω, as well as the theorem of Dalang-Morton-Willinger and KabanovRásonyi-Stricker, pertaining to the case of general Ω.An example of a 5 × 5-dimensional process (Π t ) 2 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, Rásonyi and Stricker.

Efficient portfolios in financial markets with proportional transaction costs

“…Let us recall the basic features of the transaction costs model as formalized in [4] (see also [28]). In such a model, all agents can trade in d assets according to a random and time varying bid-ask matrix.…”

“…In this paper we characterize efficient portfolios in a general multivariate financial market with transaction costs as in [28,4,3], where agents can trade in finitely many risky assets (e.g. foreign currency) facing transaction costs at each trading.…”

Arbitrage in markets with bid-ask spreads

“…A general version of this theorem was proved by Kabanov and Stricker (2001b) but in the model with finite state space Ω. Soon after, in his famous paper Schachermayer (2004) gave equivalent conditions for so-called robust no-arbitrage. The general theorem states that robust no-arbitrage is equivalent to the existence of a strictly consistent price system (CPS).…”

“…The general theorem states that robust no-arbitrage is equivalent to the existence of a strictly consistent price system (CPS). Moreover, robust no-arbitrage cannot be replaced by strict no-arbitrage due to a counter-example presented by Schachermayer (2004). Further in this direction, a very interesting and surprising theorem was proved by Grigoriev (2005) who generalised the result of Kabanov and Stricker (2001b) to arbitrary Ω in the special case of two assets.…”

Convex Duality with Transaction Costs

“…Also in all of these examples not the null sets of P but rather the support of it is important. The related question of fundamental theorem of asset pricing and super-heding duality with a given P is studied by Schachermayer [20,21] and the references therein.…”