This paper is concerned with the estimation of the risk-neutral probability distribution of a financial asset from given noisy prices of the asset and its call options. Since the number of such prices is typically much smaller than the dimension of the probability space, the problem is highly underdetermined and ill-posed. The application to this problem of linear inverse theory, which is specially suited to such underdetermined systems, is investigated in detail. Compared with other optimization methods which do not preprocess the price data, linear inverse theory shows that, for noisy input prices, significant improvement in reconstruction of a simulated distribution is obtained. The method is also applied to real market data.