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We estimate the process underlying the pricing of American options by using higher-order lattices combined with a multigrid method. This paper also tests whether the risk-neutral densities given from American options provide a good forecasting tool. We use a nonparametric test of the densities that is based on the inverse probability functions and is modified to account for correlation across time between our random variables, which are uniform under the null hypothesis. We find that the densities based on the Americanoption markets for foreign exchange do quite well for the forecasting period over which the options are thickly traded. Further, simple models that fit the densities do about as well as more sophisticated models.