The No-Arbitrage Property under a Change of Numéraire (1995)

Freddy Delbaen () and Walter Schachermayer ()
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Freddy Delbaen: ETH Zürich
Walter Schachermayer: Technische Universität Wien

Chapter 11 in The Mathematics of Arbitrage, 2006, pp 217-230 from Springer

Abstract: For a price process that has an equivalent risk neutral measure, we investigate if the same property holds when the numéraire is changed. We give necessary and sufficient conditions under which the price process of a particular asset — which should be thought of as a different currency — can be chosen as new numéraire. The result is related to the characterisation 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 numéraire. 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 generalises results previously obtained by Jacka, Ansel-Stricker and the authors.

Date: 2006
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DOI: 10.1007/978-3-540-31299-4_11

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