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The work presented here deals with several aspects of financial mathematics. A general framework is established in which the existence of stochastic processes can be guaranteed which can be used to model financial instruments. A risk neutral evaluation of such instruments is explained, while focusing on the modeling of interest forward rates, so called Libors. These allow to work without the usual assumption of the existence of a risk neutral financial instrument. Next we deal with an extended Libor market model offering enough flexibility to accurately calibrate to various market data for caplets and swaptions. To compete with standard models with regard to calibration, analytical price formulas for European options within this model are called for. Those are obtained after an approximation using the concept of Fast Fourier Transform. Further we are interested in the evaluation of more complex financial derivatives for instance by simulation. Due to the typically high dimension of the model involved such simulations can be very time consuming. We show possible improvements regarding the complexity of the simulation. Factor reduction, a method known from statistics, identifies the main driving factors of a model and may lead to a significant reduction of complexity. In addition we extend a known simulation scheme, established in Andersen [2007], from one to multiple dimensions using the concept of moment matching for the approximation of the vola process in a Heston model. This results in an improved convergence of the whole process thus reducing the complexity of a simulation based evaluation of options. Next we address the problem of evaluating so called American options. These vastly traded options can be interpreted as a stopping problem. An efficient evaluation of these options, particularly in high dimensions, is a delicate problem. For example when dealing with basket cash flows, a simulation based approach offering dimension independent convergence often happens to be the only practicable solution. A new method of variance reduction given by the multilevel idea is applied to this approach. Starting with the ideas of Belomestny and Schoenmakers [2011] a lower bound for the option price is obtained using “multilevel policy iteration” method. Convergence rates for the simulation of the option price are obtained and a detailed complexity analysis is presented. Finally we deal with the valuation of American options under model uncertainty. This lifts the restriction of considering one particular probabilistic model only. Different models might be plausible and may lead to different option values. This approach leads to a non-linear expectation functional, calling for a generalization of the standard expectation case. We obtain a generalized Snell envelope, enabling a backward recursion via Bellman principle. We then provide a numerical algorithm to valuate American options under ambiguity, obtaining lower and upper price bounds.

Stochastic volatility Libor modeling and efficient algorithms for optimal stopping problems