A General Version of the Fundamental Theorem of Asset Pricing (1994)

A basic result in mathematical finance, sometimes called the fundamental theorem of asset pricing (see [DR 87]), is that for a stochastic process $(S_{t})_{t\in \mathbb {R}_{+}}$ , the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage opportunities. In finance the process $(S_{t})_{t\in \mathbb {R}_{+}}$ describes the random evolution of the discounted price of one or several financial assets. The equivalence of no-arbitrage with the existence of an equivalent probability martingale measure is at the basis of the entire theory of “pricing by arbitrage”. Starting from the economically meaningful assumption that S does not allow arbitrage profits (different variants of this concept will be defined below), the theorem allows the probability P on the underlying probability space (Ω,ℱ,P) to be replaced by an equivalent measure Q such that the process S becomes a martingale under the new measure. This makes it possible to use the rich machinery of martingale theory. In particular the problem of fair pricing of contingent claims is reduced to taking expected values with respect to the measure Q. This method of pricing contingent claims is known to actuaries since the introduction of actuarial skills, centuries ago and known by the name of “equivalence principle”.