Modeling Anticipations on Financial Markets

Abstract

The aim of the present survey is to give an outline of the modern mathematical tools which can be used on a financial market by a ”small” investor who possesses some information on the price process.

Financial markets obviously have asymmetry of information. That is, there are different types of traders whose behavior is induced by different types of information that they possess. Let us consider a ”small” investor who trades in an arbitrage free financial market so as to maximize the expected utility of his wealth at a given time horizon. We assume that he is in the following position: He possesses extra information about some functional Y of the future prices of a stock (e.g. value of the price at a given date, hitting times of given values, ...). Our basic question is then: What is the value of this information ? We can imagine two modeling approaches:

• 1 A strong approach: The investor knows the functional ω by ω . This modeling of the additional information was initiated in [41] using initial enlargement of filtration, a theory developed in [27], [28], and [29].

• 2 A weak approach: The investor knows the law of the functional Y under the effective probability of the market assumed to be unknown. This notion of weak information is defined in [5], [6], and further studied in [8].

In this chapter, we present and compare these two approaches.

This work was begun in the ”Laboratoire de Probabilités et Modèles Aléatoires” in Paris 6 and concluded in Vienna where it was supported by the Research Training Network HPRN-CT-2002-00281. It stems partially from conferences given at the CREST in Malakoff at the beginning of year 2002. I thank Nizar Touzi and Huyen Pham for their reactions and their interesting remarks. I also thank friendly Holly for the reading of the manuscript and the correction of many faults in the use of the English language. Finally, I thank the referees for their judicious corrections and their careful reading.