A discretization scheme for nonnegative diffusion processes is proposed and the convergence of the corresponding sequence of approximate processes is proved using the martingale problem framework. Motivations for this scheme typically come from finance, especially for path-dependent option pricing. The scheme is simple: one only needs to find a nonnegative distribution whose mean and variance satisfy a simple condition to apply it. Then, for virtually any (path-dependent) payout, Monte Carlo option prices obtained from this scheme will converge to the theoretical price. Examples of models and diffusion processes for which the scheme applies are provided.