Before revision:  The absence of arbitrage opportunities is guaranteed by penalizing the loss using soft constraints on an extended grid of input values. ANNs can be pre-trained by first calibrating a standard option pricing model, and then training an ANN to a larger synthetic dataset generated from the calibrated model. The parameters transfer as well as the non-arbitrage constraints appear to be particularly useful when only sparse or erroneous data are available.
After revision:  The absence of arbitrage opportunities is guaranteed by penalizing the loss using soft constraints . Furthermore, our method can be combined with standard IVS models in quantitative finance, thus providing a NN-based correction when such models fail at replicating observed market prices. This lets practitioners use our approach as a plug-in on top of classical methods. Empirical results show that this approach is particularly useful when only sparse or erroneous data are available.