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Walter Schachermayer

The No-Arbitrage Property under a change of numéraire.

F. Delbaen, W. Schachermayer
Stochastics and Stochastic Reports, Vol. 53 (1995), pp. 213-226. [R]

Abstract:

We show that, if we allow general admissible integrands as trading strategies, the three dimensional Bessel process, Bes$^3$, admits arFor 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 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 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 generalizes results previously obtained by Jacka, Ansel-Stricker and the authors.

Keywords:

arbitrage, martingale, local martingale, equivalent martingale measure, representing measure, risk neutral measure, duality relation, hedging, stochastic integration, mathematical finance.

1991 Mathematics Subject Classification:

90A09, 60G44, 46N10, 47N10, 60H05, 60G40.

Preprints:

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Publications marked with [R] have appeared in refereed journals.

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