ThispaperisconcernedwithnumericalmethodsforAmericanoptionpricing.Weemploynumericalanalysisandthenotionofviscositysolutiontoshowuniformconvergenceoftheexplicitdifferenceschemeandthebinomialtreemethod.Wealsoprovetheexistenceandconvergenceoftheoptimalexerciseboundariesintheaboveapproximn.tions.