Before revision:  Extensions to the multivariate case, i.e. basket options and options on the minimum or maximum of several assets are considered as well. As a link to credit risk we derive a valuation formula for equity default swaps. The key idea of this Fourier, resp. Laplace transform based approach is the clear separation of two ingredients, namely the payoff function and the underlying process .
After revision:  Extensions to the multivariate case, i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the underlying process .