The finite-difference value function of an American put option can be computed by solving a sequence of linear complementarity problems (LCPs). The stateof- the-art methods that solve these equations converge slowly. Recently, Borici and Lüthi have shown that in the case of the implicit discretization scheme it is possible to solve LCPs in a computer time which grows linearly with the number of spatial grid points. In this paper we show that this result can be generalized for the more accurate discretization scheme of Crank and Nicolson. We give examples that illustrate this result.