Two essential quantities for the analysis of approximation schemes of evolution equations are stability and convergence. We derive stability and convergence of fully discrete approximation schemes of solutions to linear parabolic evolution equations governed by time-dependent coercive operators. We consider abstract Galerkin approximations in space combined with θ-schemes in time. The level of generality of our analysis comprises both a large class of time-dependent operators and a large choice of approximating Galerkin spaces. In particular, the results apply to partial integrodifferential equations for option pricing in time-inhomogeneous Lévy models and allow for a large variety of option types and models. The derivation builds on the strong foundation laid out in a 2003 paper by von Petersdorff and Schwab, which provides the respective results for the time-homogeneous case. We discuss the assumptions in the context of option pricing.