In this article, the authors reexamine the American-style option pricing formula of R. Geske and H.E. Johnson (1984), and extend the analysis by deriving a modified formula that can overcome the possibility of nonuniform convergence (which is likely to occur for nonstandard American options whose exercise boundary is discontinuous) encountered in the original Geske-Johnson methodology. Furthermore, they propose a numerical method, the Repeated-Richardson extrapolation, which allows the estimation of the interval of true option values and the determination of the number of options needed for an approximation to achieve a given desired accuracy. Using simulation results, our modified Geske-Johnson formula is shown to be more accurate than the original Geske-Johnson formula for pricing American options, especially for nonstandard American options. This study also illustrates that the Repeated-Richardson extrapolation approach can estimate the interval of true American option values extremely well. Finally, the authors investigate the possibility of combining the binomial Black-Scholes method proposed by M. Broadie and J.B. Detemple (1996) with the Repeated-Richardson extrapolation technique.