PIDE methods and concepts for parametric option pricing
This thesis is concerned with methods for option pricing that we investigate both theoretically and numerically. The first main part interprets option prices as solutions to partial integro differential equations (PIDEs) Focusing on exponential Levy models, we implement a numerical tool for solving PIDEs using a Galerkin finite element approach that is flexible in the driving asset process. Many numerical examples provide evidence for the numerical feasibility of the method. Furthermore we establish a stability and convergence analysis for PIDEs with time inhomogeneous operators of Garding type. The second part of the thesis applies Chebyshev polynomial interpolation to option pricing by interpreting option prices as functions of option and model parameters. A numerical implementation of the pricing interpolation technique illustrates the method and emphasizes the gain in efficiency. The third part combines the empirical interpolation algorithm of Barrault et al. (2004) with Fourier based option pricing by interpolating associated Fourier integrands. Theoretical findings are numerically validated. Further numerical studies highlight the appealing features of the method, especially in higher dimensional parameter spaces. Additionally, the recursive nature of the interpolation operator is resolved which renders the method numerically accessible for the interpolation of multivariate Fourier integrands, as well.