A Fundamental Theorem of Asset Pricing for Large Financial Markets

Abstract

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"-super-"n") "n"=1 -super-∞ of stochastic stock price processes based on a sequence (Ω-super-"n", "F"-super-"n", ("F" "t" -super-"n") "t" is an element of "I"-super-"n" , P-super-"n") "n"=1 -super-∞ of filtered probability spaces. Under the assumption that for all "n" is an element of N there exists an equivalent sigma-martingale measure for "S"-super-"n", we prove that there exists a "bicontiguous" sequence of equivalent sigma-martingale measures if and only if there is no asymptotic free lunch (Theorem 1.1). Moreover we present an example showing that it is not possible to improve Theorem 1.1 by replacing "no asymptotic free lunch" by some weaker condition such as "no asymptotic free lunch with bounded" or "vanishing risk." Copyright Blackwell Publishers, Inc..

No asymptotic free lunch reviewed in the light of Orlicz spaces

“…The present note is focused on the condition no asymptotic free lunch (NAFL) which is the large financial market analogue of NFL, see [11].…”

“…In general, one has to use the condition of no asymptotic free lunch (NAFL) of [11] to get this equivalence. A recent result shows that one gets the equivalence with an asymptotic condition of no market free lunch type as well, but this is a slightly different approach using preferences of investors, see [12] and [3].…”

“…In [11] the notion of no asymptotic free lunch (NAFL) was introduced and gave a general version of the Fundamental Theorem of Asset Pricing (FTAP) for large financial markets. The present note is a revision showing that the appropriate theoretical background of the rather 'hand-knitted' proof there is the theory of Orlicz spaces.…”

“…In [11] this property of bicontiguity was related to the condition NAFL (no asymptotic free lunch). The idea of NAFL is to generalize the classical no free lunch condition of Kreps [16] for a sequence of market models.…”

“…No asymptotic free lunch (NAFL) was introduced in [11] and led to a general version of the Fundamental Theorem of Asset Pricing (FTAP) for large financial markets. The present note observes that NAFL can be defined in a natural way using Orlicz spaces.…”