Existence of shadow prices in finite probability spaces

Abstract

A shadow price is a process \({\widetilde{S}}\) lying within the bid/ask prices \({\underline{S},\overline{S}}\) of a market with proportional transaction costs, such that maximizing expected utility from consumption in the frictionless market with price process \({\widetilde{S}}\) leads to the same maximal utility as in the original market with transaction costs. For finite probability spaces, this note provides an elementary proof for the existence of such a shadow price.