A HILBERT-SPACE PROOF OF THE FUNDAMENTAL THEOREM OF ASSET PRICING IN FINITE DISCRETE-TIME

Dalang, Morton and Willinger (1990) have proved a beautiful version of the Fundamental Theorem of Asset Pricing which pertains to the case of finite discrete time: In this case the absence of arbitrage opportunities already characterizes the existence of an equivalent martingale measure. The purpose of this paper is to give an elementary proof of this important theorem which relies only on orthogonality arguments. In contrast, the original proof of Dalang et al. uses heavy functional analytic machinery, in particular measurable selection and measure-decomposition theorems. We feel that the theorem (as well as its proof) should be accessible to a wider public and we therefore made an effort to keep the arguments as selfcontained as possible. In a final chapter we review and prove the necessary tools for our presentation of the theorem.