This thesis studies least-squares radial basis function partition of unity methods (LS-RBF-PUM) for the time-dependent pricing of a two-asset European arithmetic basket call. The pricing problem is formulated as a two-dimensional Black–Scholes equation on a truncated triangular domain aligned with level sets of the basket payoff variable S1 + S2. 

Two related LS-RBF-PUM formulations are considered. Method 1 uses a global set of RBF centers and serves as the main baseline solver. Method 2 uses a patch-local organization of nodal unknowns and is investigated as an alternative formulation for patch-wise refinement and non- uniform patch layouts. In both cases, the spatial discretization is constructed in an oversampled least-squares setting, while the time integration is performed in reversed time using an implicit Euler start followed by BDF2. 

In the reported Method 1 tests, the selected-point comparisons and qualitative diagnostics provide a working global-center baseline. For Method 2, the observed numerical behaviour and errors measured at five selected strike-line points vary with patch layout and boundary enforce- ment. Under the controlled structured layout, the tested configurations produce interpretable refinement and boundary diagnostics, although the spatial error remains non-monotone and the reported local condition numbers grow rapidly. Overall, the study documents the numerical be- haviour of LS-RBF-PUM for the tested time-dependent two-asset problem and identifies design factors that require further investigation. 

Faculty of Science and Technology, Uppsala University. Place of publication Uppsala. Supervisor: Elisabeth Larsson, Subject reader: Elisabeth Larsson, Examiner: Jörn Zimmerling