Option Pricing using Adaptive Meshes

Specialeforsvar ved Andreas Asp Bock

Titel: Option Pricing using Adaptive Meshes

Resume: We present a nascent adaptive finite element approach to parabolic partial differential equations modelling contingent claims with non-smooth boundary conditions. Our approach uses time-integrated systems to derive variational problems in both space and time, as opposed to semi-variational systems where space and time are discretised separately. These fully-variational problems can be solved on unstructured meshes, and we use the moving mesh framework MOM2D to treat time as a spatial dimension and allocate degrees of freedom in such a way as to abate error incurred by singularities. We use the European option as a benchmark, deriving rigorous weak formulations and reviewing classic error estimation bounds. We show that singular parabolic boundaries inhibit the usefulness of theoretical results, while numerical experiments show that adaptive finite element software can be useful in abating the error they introduce. The free boundary problem of the American option is also studied using a parametric method. We approach this problem from a fluid dynamics point of view and model the exercise boundary as an phase shift in MOM2D. Numerical results suggest that while adaptive finite element methods are ideal for modelling nonlinear interfaces they are severely taxing on the speed and therefore competitiveness of our method. Our work highlights the need for theoretical reinforcement of adaptive finite element methods for fully-variational problems and suggests future work on a posteriori error estimates and time-dependent spatial domains

Vejledere: Jens Hugger, Kenny Erleben, DIKU
Censor: Jeppe Revall Frisvad