Mesh-free methods based on radial basis function (RBF) approximation are widely used for solving PDE problems. They are flexible with respect to the problem geometry and highly accurate. A disadvantage of these methods is that the linear system to be solved becomes dense for globally supported RBFs. A remedy is to introduce localisation techniques such as partition of unity (PU). RBF-PU methods allow for significant sparsification of the linear system and lower the computational effort. In this work we apply a global RBF method as well as an RBF-PU method to problems in option pricing. We consider one- and two-dimensional vanilla options. In order to price American options we employ a penalty approach. The RBF-PU method is competitive with already existing methods and the results are promising for extension to higher-dimensional problems.
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