We present a neural-network valuation of financial derivatives in the case of fat-tailed underlying asset returns. A two-layer perceptron is trained on simulated prices taking into account the well-known effect of volatility smile. The prices of the underlier are generated using fractional calculus algorithms, and option prices are computed by means of the Bouchaud-Potters formula. This learning scheme is tested on market data; the results show a very good agreement between perceptron option prices and real market ones.