Valuing Path-Dependent Options using the Finite Element Method, Duality Techniques, and Model Reduction

Sammanfattning: In this thesis we develop an adaptive finite element method for pricing of several path-dependent options including barrier options, lookback options, and Asian options. The options are priced using the Black-Scholes PDE-model, and the resulting PDE:s are of parabolic type in one spatial dimension with different boundary conditions and jump conditions at monitoring dates. The adaptive finite element method is based on piecewise polynomial approximations in space and time. We derive a posteriori estimates for the error in pointwise values of the solution and it's derivatives, using duality techniques. The estimates are used to determine suitable local resolution in space and time. The suggested adaptive finite element method is stable and gives fast and accurate results. In addition to option prices we also calculate certain sensitivity measures, or the so called Greeks, and present a new connection between some of the Greeks and the a posteriori error analysis. We also develop an a posteriori error analysis for different SVD based model reduction techniques, and present a new model reduction technique. These techniques enables us to reduce the size of the problem, which radically improves the performance. The a posteriori error estimates are again derived using duality techniques. The model reduction techniques are tested on European and Asian options.