<p>The distribution of asset returns may lead to structural breaks. Thesebreaks may result in changes of the optimal portfolio weights. For a port-folio investor, the ability of timely detection of any systematic changesin the optimal portfolio weights is of a great interest.In this master thesis work, the use of the Shewhart method, as amethod for detecting a sudden parameter change, the implied changein the multivariate portfolio weights and its performance is reviewed.</p><p> </p>

There are some nonlinear models for pricing financial derivatives which can improve the linear Black-Scholes model introduced by Black, Scholes and Merton. In these models volatility is not constant anymore, but depends on some extra variables. It can be, for example, transaction costs, a risk from a portfolio, preferences of a large trader, etc. In this thesis we focus on these models. In the first chapter we introduce some important theory of financial derivatives. The second chapter is devoted to the volatility models. We derive three models concerning transaction costs (RAPM, Leland's  and Barles-Soner's model) and Frey's model which assumes a large (dominant) trader on the market. In the third and in the forth chapter we derive portfolio and make numerical experiments with a free boundary. We use the first order additive and the second order Strang splitting methods. We also use approximations of Barles-Soner's model using the identity function and introduce an approximation with the logarithm function of Barles-Soner's model. These models we finally compare with models where the volatility includes constant transaction costs.

The distribution of asset returns may lead to structural breaks. Thesebreaks may result in changes of the optimal portfolio weights. For a port-folio investor, the ability of timely detection of any systematic changesin the optimal portfolio weights is of a great interest.In this master thesis work, the use of the Shewhart method, as amethod for detecting a sudden parameter change, the implied changein the multivariate portfolio weights and its performance is reviewed.

Le travail de recherche décrit dans cette thèse a pour objectif d'accélérer le temps de calcul pour valoriser des instruments financiers complexes, tels des options américaines sur panier de taille réaliste (par exemple de 40 sousjacents), en tirant partie de la puissance de calcul parallèle qu'offrent les accélérateurs graphiques (Graphics Processing Units). Dans ce but, nous partons d'un travail précédent, qui avait distribué l'algorithme de valorisation de J.Picazo, basé sur des simulations de Monte Carlo et l'apprentissage automatique. Nous en proposons une adaptation pour GPU, nous permettant de diviser par 2 le temps de calcul de cette précédente version distribuée sur un cluster de 64 cœurs CPU, expérimentée pour valoriser une option américaine sur 40 actifs. Cependant, le pricing de cette option de taille réaliste nécessite quelques heures de calcul. Nous étendons donc ce premier résultat dans le but de cibler un cluster de calculateurs, hétérogènes, mixant GPUs et CPUs, via OpenCL. Ainsi, nous accélérons fortement le temps de valorisation, même si les entrainements des différentes méthodes de classification expérimentées (AdaBoost, SVM) sont centralisés et constituent donc un point de blocage. Pour y remédier, nous évaluons alors l'utilisation d'une méthode de classification distribuée, basée sur l'utilisation de forêts aléatoires, rendant ainsi notre approche extensible. La dernière partie réutilise ces deux contributions dans le cas de calcul de la Value at Risk d’un portefeuille d'options, sur cluster hybride hétérogène. / The research work described in this thesis aims at speeding up the pricing of complex financial instruments, like an American option on a realistic size basket of assets (e.g. 40) by leveraging the parallel processing power of Graphics Processing Units. To this aim, we start from a previous research work that distributed the pricing algorithm based on Monte Carlo simulation and machine learning proposed by J. Picazo. We propose an adaptation of this distributed algorithm to take advantage of a single GPU. This allows us to get performances using one single GPU comparable to those measured using a 64 cores cluster for pricing a 40-assets basket American option. Still, on this realistic-size option, the pricing requires a handful of hours. Then we extend this first contribution in order to tackle a cluster of heterogeneous devices, both GPUs and CPUs programmed in OpenCL, at once. Doing this, we are able to drastically accelerate the option pricing time, even if the various classification methods we experiment with (AdaBoost, SVM) constitute a performance bottleneck. So, we consider instead an alternate, distributable approach, based upon Random Forests which allow our approach to become more scalable. The last part reuses these two contributions to tackle the Value at Risk evaluation of a complete portfolio of financial instruments, on a heterogeneous cluster of GPUs and CPUs.