We introduce a fast and widely applicable numerical pricing method that uses recursive projections. We characterize its convergence speed. We find that the early exercise boundary of an American call option on a discrete dividend paying stock is higher under the Merton and Heston models than under the Black-Scholes model, as opposed to the continuous dividend case. A large database of call options on stocks with quarterly dividends shows that adding stochastic volatility and jumps to the Black-Scholes benchmark reduces the amount foregone by call holders failing to optimally exercise by 25%. Transaction fees cannot fully explain the suboptimal behavior.