A Fundamental Theorem of Asset Pricing for Large Financial Markets

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 (Sⁿ)_n=1^∞ of stochastic stock price processes based on a sequence (Ωⁿ, Fⁿ, (F_tⁿ)_t∈In, Pⁿ)_n=1^∞ of filtered probability spaces. Under the assumption that for all n∈ N there exists an equivalent sigma‐martingale measure for Sⁿ, 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.”