We present a novel method for deriving tight Monte Carlo confidence intervals for solutions of stochastic dynamic programming equations. Taking some approximate solution to the equation as an input, we construct pathwise recursions with a known bias. Suitably coupling the recursions for lower and upper bounds ensures that the method is applicable even when the dynamic program does not satisfy a comparison principle. We apply our method to three nonlinear option pricing problems, pricing under bilateral counterparty risk, under uncertain volatility, and under negotiated collateralization.