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
of stochastic stock price
processes based on a sequence of filtered probability spaces. Under the
assumption that, for all 
, there exists an equivalent (sigma-)
martingale measure for 
, 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 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.
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