We present a neural network (NN) approach to fit and predict implied volatility surfaces (IVSs). Atypically to standard NN applications, financial industry practitioners use such models equally to replicate market prices and to value other financial instruments. In other words, low training losses are as important as generalization capabilities. Importantly, IVS models need to generate realistic arbitrage-free option prices, meaning that no portfolio can lead to risk-free profits. We propose an approach guaranteeing the absence of arbitrage opportunities 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.

Additional Feedback: \section*{Comments} \subsection*{General comments} From a theoretical standpoint, the two novelties that the authors claim to introduce -combining a prior model with a neural network and using soft constraints for the no arbitrage conditions, are of critical importance and make this manuscript worthy of a publication to me. From a practical standpoint, the implementation details -whether regarding the loss function, the neural network inputs or the training set, could use more work before producing a useful implied volatility surface superior to the ones produced by more classical techniques. A comparison with these techniques would have been most welcome. They should at least weight those observations by: 1/The bid/ask spread (no need to fit perfectly when the market is wide, and focus on ATM options which are usually tighter than the wings) and 2/ The vega' \frac{\partial p}{\partial \sigma} of each options, which penalizes errors on options which are highly sensitive to \sigma more than ones which are less sensitive to it. Using an equivalent condition in price space might make it easier to write a more realistic version.

We present a neural network (NN) approach to fit and predict implied volatility surfaces (IVSs). Atypically to standard NN applications, financial industry practitioners use such models equally to replicate market prices and to value other financial instruments. In other words, low training losses are as important as generalization capabilities. Importantly, IVS models need to generate realistic arbitrage-free option prices, meaning that no portfolio can lead to risk-free profits. We propose an approach guaranteeing the absence of arbitrage opportunities by penalizing the loss using soft constraints.