2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210162The paper considers trading with proportional transaction costs. We give a necessary and sufficient condition for A, the cone of claims attainable from zero endowment, to be closed, and show, in general, how to represent its closure in such a way that it is the cone of claims attainable for zero endowment, for a different set of trading prices. The new representation obeys the Fundamental Theorem of Asset Pricing. We then show how to represent claims and in a final section show how any such setup corresponds to a coherent risk measure.15 pages. Contains simplifications and amended notationProbabilityTrading and Market Microstructure91B24, 91B28, 52A20text