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978-3-8439-0394-3, Reihe Informatik
A Pricing Framework for the Efficient Evaluation of Financial Derivatives based on Theta Calculus and Adaptive Sparse Grids
280 Seiten, Dissertation Technische Universität München (2012), Softcover, B5
Due to the growing complexity and relevance of financial products, computational finance has recently become an emerging discipline. Sophisticated mathematical models are used to develop and validate new financial products. From this, also higher-dimensional pricing problems result, which are extremely challenging to solve as they have to be treated in a computationally highly efficient way. The increasing variety of the financial contracts further demands for more flexible pricing frameworks which allow to tackle different problems with one software product in order to avoid code redundancy and re-development.
Up to now, often, Monte Carlo methods are used to solve the resulting pricing problems, but they exhibit a slow convergence. For obtaining higher accuracy, PDE-based pricing methods are preferred. Unfortunately, PDE approaches suffer from the curse of dimensionality. Furthermore, due to their complexity, current PDE approaches to financial pricing problems usually treat only a single, specific pricing problem. Consequently, there is a need for both efficient numerics and a direct and automated path from modeling financial products to their numerical simulation.
In this thesis, a general financial pricing framework has been developed. The domain-specific modeling language ThetaML was used to model the structure of a financial product with a formal calculus. Based on this representation, a numerical simulation is performed to evaluate the product. The underlying PDE-based pricing algorithm is flexible and supports different stochastic models involving Brownian processes as well as mean-reverting processes. For the discretization, spatially adaptive sparse grids were employed in order to deal also with higher dimensions. With the implemented pricing toolbox, it is now possible for the first time to price a variety of financial products in up to six dimensions PDE-based within one pricing framework. Products that can be evaluated cover European and American multi-asset options, different exotic options, and bond derivatives, but also life insurance products in form of variable annuities.