We develop a simulation algorithm for estimating the prices of American-style securities, i.e., securities with opportunities for early exercise. Our algorithm provides both point estimates and error bounds for the true security price. It generates two estimates, one biased high and one biased low, both asymptotically unbiased and converging to the true price. Combining the two estimators yields a confidence interval for the true price. The proposed algorithm is especially attractive (compared with lattice and finite-difference methods) when there are multiple state variables and a small number of exercise opportunities. Preliminary computational evidence is given.