We consider the Heston model as an example of a parameterized parabolic partial differential equation. A space-time variational formulation is derived that allows for parameters in the coefficients (for calibration) and enables us to choose the initial condition (for option pricing) as a parameter function. A corresponding discretization in space and time for the initial condition are introduced. Finally, we present a novel reduced basis method that is able to use the initial condition of the parabolic partial differential equation as a parameter (function). The corresponding numerical results are shown.