Analysis and numerical methods for stochastic volatility models in valuation of financial derivatives

espanolEl objetivo principal de la tesis se centra en el estudio del modelo de volatilidad estocastica SABR para los subyacentes (activos o tipos de interes) con vista a la valoracion de diferentes productos derivados. En el caso de los derivados de tipos de interes, el modelo SABR se combina con el modelo de mercado de tipos de interes mas popular en estos momentos, el LIBOR market model (LMM). Los metodos numericos de valoracion son fundamentalmente de tipo Monte Carlo y la resolucion numerica de los modelos de ecuaciones en derivadas parciales (EDPs) correspondientes. Las EDPs asociadas a modelos SABR/LIBOR tienen alta dimension en espacio, por lo que se estudian tecnicas de sparse grid para vencer la maldicion de la dimension. Ademas, se discute detalladamente como calibrar los parametros de los modelos a las cotizaciones de mercado, para lo cual se propone el uso del algoritmo de optimizacion global estocastica Simulated Annealing. Los algoritmos citados tienen un alto coste computacional. Con el objetivo de que tanto las valoraciones como las calibraciones se hagan en el menor tiempo posible se emplean diferentes tecnicas de computacion de altas prestaciones (multicomputadores, multiprocesadores y GPUs.) Finalmente se dise~na un nuevo algoritmo basado en Least-Squares Monte Carlo (LSMC) para aproximar la solucion de Backward Stochastic Differential Equations (BSDEs). EnglishThe main objective of this thesis concerns to the study of the SABR stochastic volatility model for the underlyings (equity or interest rates) in order to price several market derivatives. When dealing with interest rate derivatives the SABR model is joined with the LIBOR market model (LMM) which is the most popular interest rate model in our days. In order to price derivatives we take advantage not only of Monte Carlo algorithms but also of the numerical resolution of the partial di erential equations (PDEs) associated with these models. The PDEs related to SABR/LIBOR market models are high dimensional in space. In order to cope with the curse of dimensionality we will take advantage of sparse grids. Furthermore, a detailed discussion about the calibration of the parameters of these models to market prices is included. To this end the Simulated Annealing global stochastic minimization algorithm is proposed. The above mentioned algorithms involve a high computational cost. In order to price derivatives and calibrate the models as soon as possible we will make use of high performance computing (HPC) techniques (multicomputers, multiprocessors and GPUs). Finally, we design a novel algorithm based on Least-Squares Monte Carlo (LSMC) in order to approximate the solution of Backward Stochastic Di erential Equations (BSDEs).