In the international financial markets option products are widely traded. Advanced mathematical models are employed for determining the fair values of these contracts. This leads to multidimensional time-dependent partial differential equations (PDEs). For the majority of these PDEs there is no analytical solution available and one resorts to numerical methods for their approximate solution. A well-known and versatile approach to the numerical solution is given by the method-of-lines. The PDE is then first discretized in the spatial variables, leading to a large system of ordinary differential equations. In a second step this semidiscrete system is numerically solved by applying a suitable implicit time discretization method. If the PDE is multidimensional, then the latter task can be computationally intensive when classical implicit time stepping methods are used.

In this thesis we consider the convergence and application of four alternating direction implicit (ADI) time stepping schemes in the numerical solution of semidiscretized two-dimensional convection-diffusion equations. More precisely, we consider the Douglas scheme, the Craig-Sneyd (CS) scheme, the Modified Craig-Sneyd (MCS) scheme and the Hundsdorfer-Verwer (HV) scheme. ADI schemes employ a splitting of the semidiscrete PDE operator in the different spatial directions. This can lead to a major computational advantage in each time step as it turns out that the implicitness is often much easier to deal with when the suboperators are handled successively, instead of treating the full operator all at once.

We prove that, under natural stability and smoothness assumptions, the (M)CS scheme and HV scheme are second order convergent uniformly in the arbitrarily small spatial mesh width. If the initial function is non-smooth, then application of the ADI schemes can lead to spurious erratic behaviour of the numerical solution. It is shown that by applying Rannacher time stepping, the classical order of convergence can be recovered for the four ADI schemes. Finally, we introduce two methods for the calibration of state-of-the-art stochastic local volatility models. Here, the ADI schemes contribute to the speed, stability and accuracy of the calibration procedures.