Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

Auteur(s):

Code(s) de Classification MSC:

• 91B28 Finance, portfolios, investment
• 60G48 Generalizations of martingales
• 91B70 Stochastic models

Résumé: We consider the hedging problem in an arbitrage-free financial market, where there are two kinds of investors with different levels of information about the future price evolution, described by two filtrations $\mathbf F$ and $\mathbf G =\mathbf F \vee \sigma (G)$ where $G$ is a given r.v. representing the additional information. We focus on two types of quadratic approaches to hedge a given square-integrable contingent claim: local risk minimization (LRM) and mean-variance hedging (MVH). By using initial enlargement of filtrations techniques, we solve the hedging problem for both investors and compare their optimal strategies under both approaches. In particular, for LRM, we show that for a large class of additional non trivial r.v.s $G$ both investors will pursue the same locally risk minimizing portfolio strategy and the cost process of the ordinary agent is just the projection on $\mathbf F$ of that of the insider. In the MVH setting, we study also some general stochastic volatility model, including Hull and White, Heston and Stein and Stein models. In this more specific setting and for r.v.s $G$ which are measurable with respect to the filtration generated by the volatility process, we obtain an expression for the insider optimal strategy in terms of the ordinary agent optimal strategy plus a process admitting a simple backward-type representation.

Mots Clés: insider trading ; initial enlargement of filtrations ; martingale preserving measure ; local risk minimization ; mean-variance hedging ; stochastic volatility models

Date: 2003-09-09

Prépublication numéro: PMA-841

Revised version: PMA-841bis.pdf (2003-12-16)