The No - Arbitrage Property under a Change of Num eraire

For a price process that has an equivalent risk neutral measure, we investigate if the same property holds when the num eraire is changed. We give necessary and su cient conditions under which the price process of a particular asset which should be thought of as a di erent currency can be chosen as new num eraire. The result is related to the characterization of attainable claims that can be hedged. Roughly speaking: the asset representing the new currency is a reasonable investment (in terms of the old currency) if and only if the market does not permit arbitrage opportunities in terms of the new currency as num eraire. This rough but economically meaningful idea is given a precise content in this paper. The main ingredients are a duality relation as well as a result on maximal elements. The paper also generalizes results previously obtained by Jacka, Ansel-Stricker and the authors.