Iterative methods for option pricing in Merton's Jump diffusion model

The main purpose of this thesis is to study numerical schemes for the solution of the PDE associated with the Merton jump-diffusion model. The implementation of these schemes will be achieved through the finite differences method, and in particular two implicit methods (Backward Euler and Crank-Nicolson) will be developed. The linear systems created will be solved by two iterative methods, namely Multigrid and GMRES; for the latter, the effectiveness of using some preconditioners will be tested. Moreover, an explicit-implicit scheme will be applied with the Backward Euler method and it will be resolved by the direct method Tridiagonal solver

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

[Abstract] The 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).[Resumen] El objetivo principal de la tesis se centra en el estudio del modelo de volatilidad estocástica SABR para los subyacentes (activos o tipos de interés) con vista a la valoración de diferentes productos derivados. En el caso de los derivados de tipos de interés, el modelo SABR se combina con el modelo de mercado de tipos de interés más popular en estos momentos, el LIBOR market model (LMM). Los métodos numéricos de valoración son fundamentalmente de tipo Monte Carlo y la resolución numérica de los modelos de ecuaciones en derivadas parciales (EDPs) correspondientes. Las EDPs asociadas a modelos SABR/LIBOR tienen alta dimensión en espacio, por lo que se estudian técnicas de sparse grid para vencer la maldición de la dimensión. Además, se discute detalladamente cómo calibrar los parámetros de los modelos a las cotizaciones de mercado, para lo cual se propone el uso del algoritmo de optimización global estocástica 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 técnicas de computación de altas prestaciones (multicomputadores, multiprocesadores y GPUs.) Finalmente se dise~na un nuevo algoritmo basado en Least-Squares Monte Carlo (LSMC) para aproximar la solución de Backward Stochastic Differential Equations (BSDEs).[Resumo] O obxectivo principal da tese céntrase no estudo do modelo de volatilidade estocástica SABR para os subxacentes (activos ou tipos de xuro) con vista á valoración de diferentes produtos derivados. No caso dos derivados de tipos de xuro, o modelo SABR combínase co modelo de mercado de tipos de xuro máis popular nestos momentos, o LIBOR market model (LMM). Os métodos numéricos de valoración son fundamentalmente de tipo Monte Carlo e a resolución numérica dos modelos de ecuacións en derivadas parciais (EDPs) correspondentes. As EDPs asociadas aos modelos SABR/LIBOR te~nen alta dimensión en espazo, polo que se estudan técnicas de sparse grid para vencer a maldición da dimensión. Ademais, discútese detalladamente como calibrar os parámetros dos modelos ás cotizacións de mercado, para o cal se propón o emprego do algoritmo de optimización global estocástica Simulated Annealing. Os algoritmos citados te~nen un alto custo computacional. Co obxectivo de que tanto as valoracións como as calibracións se fagan no menor tempo posible empréganse diferentes técnicas de computación de altas prestacións (multicomputadores, multiprocesadores e GPUs.) Finalmente deséñase un novo algoritmo baseado en Least-Squares Monte Carlo (LSMC) para aproximar a solución de Backward Stochastic Differential Equations (BSDEs)

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators