Le propos de cette session est de présenter différents aspects tant théoriques que numériques et pratiques de la recherche actuelle en mathématiques financières.

La crise récente amène naturellement à étudier de façon approfondie la modélisation sous-jacente. Damien Lamberton abordera le problème de valorisation d’options dépendant de la trajectoire d’un actif dont la dynamique est guidée par un processus à sauts. Ying Jiao proposera quant à elle un travail relatif à la maximisation d’utilité dans le cadre d’un modèle avec délit d’initié, i.e. lorsqu’un investisseur à plus d’informations que les autres sur un certain marché.

Une fois le modèle financier donné, une autre question cruciale concerne la couverture de produits qui en dérivent. Adrien Nguyen Huu abordera les aspects spécifiquement liés aux marchés de l’électricité. Un autre aspect est celui de la gestion des risques. Noufel Frikha présentera des travaux permettant de contrôler de façon non asymptotique les déviations entre un algorithme stochastique de type Robbins-Monro et sa cible. De tels algorithmes interviennent de façon naturelle pour l’estimation de quantités telles que la Var/CVar (Value et Risk et Conditional Value at Risk) qui restent des indicateurs essentiels pour le contrôle des pertes.

Enfin, Stefano De Marco présentera quelques applications de techniques de grandes déviations pour obtenir des asymptotiques de prix d’options à grands strikes pour certains modèles à volatilité stochastique.

This talk will survey some recent results on American options, barrier and lookback options. Most of these results have been obtained in joint work with Mohammed Mikou, El Hadj Aly Dia and Ayech Bouselmi.

We consider an insider maximizing her expected utility with portfolio terminal wealth based on extra information flow compared to a standard investor. We prove that if there is no limit for short-selling strategy, then the insider can achieve unbounded utility expectation. So we propose suitable strategy constraint under which we study the optimization problem. This is a joint work with C. Hillairet.

Motivated by Gyongy’s (86) projection result for Itô SDEs and by its applications in the context of volatility modeling in finance, we discuss some large deviation results for densities of diffusions in a small-noise regime, focusing on marginals (i.e. the densities of components of the process) and on the law of the diffusion conditioned to be in some affine subspace at final time. For stochastic volatility models enjoying certain scaling properties (i.e. Stein-Stein model), this eventually leads to the large-strike asymptotic behavior of their “equivalent" local volatility function. The asymptotic value can be used to patch the typical numerical issues affecting the local volatility extracted from Dupire’s formula.

We obtain non asymptotic concentration bounds for two kinds of stochastic approximations. We first consider the deviations between the expectation of a given function of the Euler scheme of some diffusion process at a fixed deterministic time and its empirical mean obtained by the Monte-Carlo procedure. We then give some estimates concerning the deviation between the value at a given time-step of a stochastic approximation algorithm and its target. Under suitable assumptions both concentration bounds turn out to be Gaussian. The key tool consists in exploiting accurately the concentration properties of the increments of the schemes. No specific non-degeneracy conditions are assumed.

We start from the results of Bouchard and al. (Stochastic target problem with controlled loss, 2009) which provide a viscosity characterization of the value function of the following problem:

The function v defines the smallest amount y such that one can find a strategy ν for which the state variable (X_t,x ^ν, Y_t,x,y ^ν)(T) satisfies the expected constraint E[Ψ(X_t,x ^ν(T), Y_t,x,y ^ν(T))] ≥ p. In our setting, the function Ψ is a loss function applied to the portfolio Y_t,x,y endowed with a claim with payoff g(X_t,x (T)).

We recall in a first step the explicit solution provided in the complete market case, see Bouchard and al (2009). Using duality arguments, the explicit formulation provides an expectation formulation with a penalty function, as in the study of risk measures. We then extend the framework to the semi-complete framework, where non-hedgeable sources of risk appears at deterministic dates. We apply a problem reduction to retrieve a piecewise complete market case. We provides a couple of examples and several intuitions based on Bouchard and Dang (Optimal control Vs Stochastic Target : an equivalence result, 2011). Finally, since we are confronted to a non-explicit reduced formulation of the problem, we suggest numerical methods to deal with the problem. We apply the presented methods to the problem of hedging and pricing an option on a non-yet quoted futures in electricity markets.

This is joint work with Nadia Oudjane, EDF R&D.