A new and rigorously justifiable method for reducing the bias inherent in Monte Carlo estimators of American contingent claim prices is presented in this paper. This technique is demonstrated in the context of stochastic-tree estimators, not because these estimators are computationally efficient, but because they are simple enough for rigorous convergence results to be available. Large-sample theory is used to derive an easily evaluated approximation of the bias that holds for general asset-price processes of any dimensionality and for general payoff structures. This method constructs bias-corrected estimators by subtracting the bias approximation from the uncorrected estimators at each exercise opportunity. Using a well-studied multivariate pricing problem it is shown that the bias-corrected estimators significantly outperform their uncorrected counterparts across all combinations of a number of exercise opportunities, option moneyness and sample size. Furthermore, it is shown that this method is superior to a bootstrap approach for reducing bias.