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Stock options are priced numerically using space- and time-adaptive finite difference methods. European options on one and several underlying assets are considered. These are priced with adaptive numerical algorithms including a second order method and a more accurate method. For American options we use the adaptive technique to price options on one stock with and without stochastic volatility. In all these methods emphasis is put on the control of errors to fulfill predefined tolerance levels. The adaptive second order method is compared to an alternative discretization technique using radial basis functions. This method is not adaptive but shows potential in option pricing for one and several underlying assets. A finite difference method and a Monte Carlo method are applied to a new financial contract called Turbo warrant. A comparison of these two methods shows that for the case considered the finite difference method is superior.

Radial basis function (RBF) methods are meshfree, i.e., they can operate on unstructured node sets. Because the only geometric information required is the pairwise distance between the node points, these methods are highly flexible with respect to the geometry of the computational domain. The RBF approximant is a linear combination of translates of a radial function, and for PDEs the coefficients are found by applying the PDE operator to the approximant and collocating with the right hand side data. Infinitely smooth RBFs typically result in exponential convergence for smooth data, and they also have a shape parameter that determines how flat or peaked they are, and that can be used for accuracy optimization. In this thesis the focus is on global RBF collocation methods for PDEs, i.e., methods where the approximant is constructed over the whole domain at once, rather than built from several local approximations. A drawback of these methods is that they produce dense matrices that also tend to be ill-conditioned for the shape parameter range that might otherwise be optimal. One current trend is therefore to use over-determined systems and least squares approximations as this improves stability and accuracy. Another trend is to use localized RBF methods as these result in sparse matrices while maintaining a high accuracy. Global RBF collocation methods together with RBF interpolation methods, however, form the foundation for these other versions of RBF--PDE methods. Hence, understanding the behaviour and practical aspects of global collocation is still important. In this thesis an overview of global RBF collocation methods is presented, focusing on different versions of global collocation as well as on method properties such as error and convergence behaviour, approximation behaviour in the small shape parameter range, and practical aspects including how to distribute the nodes and choose the shape parameter value. Our own research illustrates these different aspects of global RBF collocation when applied to the Helmholtz equation and the Black-Scholes equation.