We introduce a new numerical approach, called the “singular points method”, for pricing American path-dependent options. This method, which is based on a continuous representation of the price at each node of the binomial tree, allows us to obtain very precise upper and lower bounds for the discrete binomial price. Moreover, the method provides a priori estimates of the difference between the upper and lower bounds. The algorithm is convergent and provides efficient estimates of the continuous price value. We apply the method to the case of Asian and lookback American options.