Existence of price functionals with bid-ask spreads on the space of all integrable contingent claims

Jouini [Price functionals with bid-ask spreads: an axiomatic ap-proach, Journal of Mathematical Economics 34 (2000), 547-558] presented an axiomatization of the existence of an admissible price functional in case that the set of marketed contingent claims consists of all square integrable random variables. The author characterized the absence of arbitrage opportunities by requiring the admissible price functional to be strictly positive. In this paper, we present a sufficient condition for the existence of an admissible price functional in the presence of bid-ask spreads in the more general situation when the space of marketed contingent claims is equal to the space of all integrable contingent claims. The assumption of strict positivity of an admissible price functional is replaced by the stronger assumption of strict monotonicity with respect to the natural preorder on the space of all integrable contingent claims. Our result generalizes the sufficiency part in the axiomatization proposed by Jouini in the aforementioned paper.