The Fundamental Theorem of Asset Pricing for Liquid Financial Markets

Abstract

A financial market is said to be liquid if each contingent claim can be traded at every time. Let E be the evolution of asset prices and F a general flow of information that encompasses E . The market is said to be sensitive to F if E “fully reflects” or “rapidly adjusts to” the information flow F . The exact meaning of “fully reflects” and “rapidly adjusts to” is specified in terms of martingale theory and it is demonstrated that market sensitivity is intimately connected to different notions of the Efficient-Market Hypothesis. Further, the market is said to be arbitrage free if there is no dominance and no free lunch with vanishing risk. Let P be the real-world probability measure. It is shown that every liquid financial market contains a numeraire asset such that each discounted price process is a P-martingale with respect to F if and only if it is arbitrage free and sensitive to F . Moreover, the numeraire asset is growth optimal on F and unique. Hence, if the financial market is arbitrage free, sensitive, and liquid, risk-neutral and real-world valuation coincide. This is a fundamental theorem of asset pricing and leads to a substantial generalization of Samuelson’s martingale hypothesis.