We consider the valuation of derivative securities by variational methods. In particular, the expectations of such processes may be represented as solutions of variational inequalities of evolutionary type typically characterized by multiple time variables, nonlocal behavior, high number of degrees of freedom, unbounded state spaces, and lack of boundary conditions. We introduce a general framework for such valuations as well as Galerkin methodologies for their constructive approximations. Results are implemented utilizing finite elements; benchmarks are provided which validate the applicability and efficiency of the method and include the valuation of options on multiple assets, options depending on path history, and options on a foreign currency.