We consider the pricing of a special kind of option, the so-called autocallable, which may terminate prior to maturity due to a barrier condition on one or several underlyings. Standard Monte Carlo algorithms work well for pricing these options but they do not behave stably with respect to numerical differentiation. Hence, to calculate sensitivities, we would typically resort to regularized differentiation schemes or derive an algorithm for directly calculating the derivative. In this work, we present an alternative solution and show how to adapt a Monte Carlo algorithm in such a way that its results can be stably differentiated by simple finite differences. Our main tool is the one-step survival idea of Glasserman and Staum, which we combine with a technique known as GHK importance sampling for treating multiple underlyings. Besides the stability with respect to differentiation, our new algorithm also possesses a significantly reduced variance and does not require evaluations of multivariate cumulative normal distributions.