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The notion of arbitrage is crucial to the modern theory of Finance. It is the corner-stone of the option pricing theory due to F. Black, R. Merton and M. Scholes [BS 73], [M 73] (published in 1973, honoured by the Nobel prize in Economics 1997).

In this section we shall develop the theory of pricing and hedging of derivative securities in financial markets.

In addition to the model of a financial market, we now consider a function (), modelling the utility of an agent’s wealth at the terminal time .

In this chapter we illustrate the theory developed in the previous chapters by analyzing the most basic examples in continuous time. They still play an important role in practice.

Let us turn back to the no-arbitrage theory developed in Chap. 2 to raise again the question: what can we deduce from applying the no-arbitrage principle with respect to pricing and hedging of derivative securities?

In Chap. 2 we only dealt with finite probability spaces. This was mainly done because of technical difficulties. As soon as the probability space (,ℱ,) is no longer finite, the corresponding function spaces such as (,ℱ,) or (,ℱ,) are infinite dimensional and we have to fall back on functional analysis. In this chapter we will present a proof of the Fundamental Theorem of Asset Pricing, Theorem 2.2.7, in the case of general (,ℱ,), but still in finite discrete time. Since discounting does not present any difficulty, we will suppose that the -dimensional price process has already been discounted as in Sect. 2.1. Also the notion of the class ℋ of trading strategies does not present any difficulties and we may adopt Definition 2.1.4 verbatim also for general (,ℱ,) as long as we are working in finite discrete time. In this setting we can state the following beautiful version of the Fundamental Theorem of Asset Pricing, due to Dalang, Morton and Willinger [DMW 90].

In the previous chapters we mainly developed the arbitrage theory for models in finite discrete time. In the setting of the previous chapter, where the probability space was not finite, several features of infinite dimensional functional analysis played a role. When trading takes place in continuous time the difficulties increase even more. It is here that we need the full power of stochastic integration theory. Before giving precise definitions, let us give a short overview of the different models and of their mutual relation.

After all this preliminary work we are finally in a position to tackle the theme of no-arbitrage in full generality, i.e., for general models of financial markets in continuous time, and for general (i.e., not necessarily simple) trading strategies . The choice of the proper class of trading strategies will turn out to be rather subtle. In fact, for different applications (e.g., portfolio optimisation with respect to exponential utility to give a concrete example; see [DGRSSS 02] and [S 03a]) it will sometimes be necessary to consider different classes of appropriate trading strategies. But for the present purpose the concept of strategies developed below will serve very well.

A basic result in mathematical finance, sometimes called the (see [DR 87]), is that for a stochastic process , the existence of an equivalent martingale measure is equivalent to the absence of arbitrage opportunities. In finance the process 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 does not allow arbitrage profits (different variants of this concept will be defined below), the theorem allows the probability on the underlying probability space (,ℱ,) to be replaced by an equivalent measure such that the process 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 . 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”.

We give an easy example of two strictly positive local martingales which fail to be uniformly integrable, but such that their product is a uniformly integrable martingale. The example simplifies an earlier example given by the second author. We give applications in Mathematical Finance and we show that the phenomenon is present in many incomplete markets.