Infinitely many securities and the fundamental theorem of asset pricing

Abstract

Several authors have pointed out the possible absence of martingale measures for static arbitrage free markets with an infinite number of available securities. Accordingly, the literature constructs martingale measures by generalizing the concept of arbitrage (free lunch, free lunch with bounded risk, etc.) or introducing the theory of large financial markets. This paper does not modify the definition of arbitrage and addresses the caveat by drawing on projective systems of probability measures. Thus we analyze those situations for which one can provide a projective system of σ−additive measures whose projective limit may be interpreted as a risk-neutral probability of an arbitrage free market. Hence the Fundamental Theorem of Asset Pricing is extended so that it can apply for models with infinitely many assets.