Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**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)