In this paper, we develop a Monte Carlo method that enables us to price instruments with discontinuous payoffs and nonsmooth trigger functions; this allows a stable computation of Greeks via finite differences. The method extends the idea of smoothing the payoff to the multivariate case. This is accomplished by a coordinate transformation and a one-dimensional analytic treatment with respect to the locally most important coordinate. It also entails Monte Carlo sampling with respect to other coordinates. In contrast with other approaches, our method does not use importance sampling. This allows us to reuse simulated paths for the pricing of other instruments or for the computation of finite-difference Greeks, something which leads to massive savings in computational cost. Avoiding the use of importance sampling leads to a certain bias, which is usually very small. We give a numerical analysis of this bias and show that simple local time grid refinement is sufficient to always keep the bias within low limits. Numerical experiments show that our method gives stable finite-difference Greeks, even in situations where payoff discontinuities occur close to the valuation date.