Many American option pricing models can be formulated as linear complementarity problems (LCPs) involving partial differential operators. While recent work with this approach has mainly addressed the model classes where the resulting LCPs are highly structured and can be solved fairly easily, this paper discusses a variety of option pricing models that are formulated as partial differential complementarity problems (PDCPs) of the convection-diffusion kind whose numerical solution depends on a better understanding of LCP methods. Specifically, the authors present second-order upwind finite-difference schemes for the PDCPs and derive fundamental properties of the resulting discretized LCPs that are essential for the convergence and stability of the finite-difference schemes and for the numerical solution of the LCPs by effective computational methods. Numerical results are reported to support the benefits of the proposed schemes. A main objective of this presentation is to elucidate the important role that the LCP has to play in the fast and effective numerical pricing of American options.