Given standard diffusion-based option pricing assumptions and a set of traded European option quotes and their payoffs at maturity, the authors identify a unique and stable set of diffusion coefficients or volatilities. Effectively, a set of option prices are inverted into a state- and time-dependent volatility function. The problem differs from the standard direct problem in which volatilities and maturity payoffs are known and the associated option values are calculated. Specifically, the authors' approach, which is based on a small-parameter expansion of the option value function, is a finite difference based procedure. It builds on previous work, which has followed Tikhonov's treatment of integral equations on the Fredholm or convolution type. An implementation of this approach with CBOE S&P500 option data is also discussed.