Standard Monte Carlo methods can often be significantly improved with the addition of appropriate variance reduction techniques. In this paper a new and powerful variance reduction technique is presented. The method is based directly on the It calculus and is used to find unbiased variance-reduced estimators for the expectation of functionals of It diffusion processes. The approach considered has wide applicability: for instance, it can be used as a means of approximating solutions of parabolic partial differential equations or applied to valuation problems that arise in mathematical finance. We illustrate how the method can be applied by considering the pricing of European-style derivative securities for a class of stochastic volatility models, including the Heston model.