15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 10 - Angewandte Mathematik, Industrie- und Finanzmathematik
Montag, 17. September 2001, 18.00, Hörsaal 42


Large Financial Markets and Asymptotic Free Lunch

Irene Klein, Universität Wien


We formulate the notion of asymptotic free lunch which is closely related to the condition ``free lunch'' of Kreps (1981) and allows us to state and prove a fairly general version of the fundamental theorem of asset pricing in the context of a large financial market as introduced by Kabanov and Kramkov (1994). In a large financial market one considers a sequence $ (S^n)_{n=1}^{\infty}$ of stochastic stock price processes based on a sequence of filtered probability spaces. Under the assumption that, for all $ n$, there exists an equivalent (sigma-) martingale measure for $ S^n$, we prove that there exists a bicontiguous sequence of equivalent (sigma-) martingale measures if and only if there is no asymptotic free lunch. Moreover there is an example showing that, in general, it is not possible to improve the result by replacing the rather technical notion no asymptotic free lunch by some weaker and economically more reasonable condition such as no asymptotic free lunch with bounded or vanishing risk. However, if we additionally assume that the processes $ S^n$ are continuous, for all n, then no asymptotic free lunch with bounded risk is necessary and sufficient for the existence of a sequence of local martingale measures, that is, in this case we obtain a version of the fundamental theorem of asset pricing, that allows a more satisfying economic interpretation.

E-Mail: Irene.Klein@univie.ac.at

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