The Fundamental Theorem of Asset Pricing

The story of this theorem started like most of modern Mathematical Finance with the work of F. Black, M. Scholes [3] and R. Merton [25]. These authors consider a model S = (St)0≤t≤T of geometric Brownian motion proposed by P. Samuelson [30], which today is widely known under the name of Black–Scholes model. Presumably every reader of this article is familiar with the by now wellknown technique to price options in this framework (compare eqf04/003: Risk Neutral Pricing): one changes the underlying measure P to an equivalent measure Q under which the discounted stock price process is a martingale. Subsequently one prices options (and other derivatives) by simply taking expectations with respect to this “risk neutral” or “martingale” measure Q. In fact, this technique was not the novel feature of [3] and [25]. It was used by actuaries for some centuries and it was also used by L. Bachelier [2] in 1900 who considered Brownian motion (which, of course, is a martingale) as a model S = (St)0≤t≤T of a stock price process. In fact, the prices obtained by Bachelier by this method were at least for the empirical data considered by Bachelier himself very close to those derived from the celebrated Black– Merton–Scholes formula (compare [34]). The decisive novel feature of the Black–Merton–Scholes approach was the argument which links this pricing technique with the notion of arbitrage: the pay-off function of an option can be precisely replicated by trading dynamically in the underlying stock. This idea, which is credited in footnote 3 of