Price functionals with bid–ask spreads: an axiomatic approach

In Jouini and Kallal [Jouini, E., Kallal, H., 1995. Martinagles and arbitrage in securities markets with transaction costs. Journal of Economic Theory 66 (1) 178-197], the authors characterized the absence of arbitrage opportunities for contingent claims with cash delivery in the presence of bid–ask spreads. Other authors obtained similar results for amore general definition of the contingent claims but assuming some specific price processes and transaction costs rather than bid–ask spreadsin general (see for instance, Cvitanic and Karatzas [Cvitanic, J., Karatzas, I., 1996. Hedging andportfolio optimization under transaction costs: a martinangle approach. Mathematical Finance 6,133-166]). The main difference consists of the fact that the bid–ask ratio is constant in this lastreference. This assumption does not permit to encompass situations where the prices are determinedby the buying and selling limit orders or by a (resp. competitive) specialist (resp. market-makers).Wederive in this paper some implications from the no-arbitrage assumption on the price functionalsthat generalizes all the previous results in a very general setting. Indeed, under some minimalassumptions on the price functional, we prove that the prices of the contingent claims are necessarilyin some minimal interval. This result opens the way to many empirical analyses