  Deep learning for option pricing has emerged as a novel methodology for fast
computations with applications in calibration and computation of Greeks.
However, many of these approaches do not enforce any no-arbitrage conditions,
and the subsequent local volatility surface is never considered. In this
article, we develop a deep learning approach for interpolation of European
vanilla option prices which jointly yields the full surface of local
volatilities. We demonstrate the modification of the loss function or the feed
forward network architecture to enforce (hard constraints approach) or favor
(soft constraints approach) the no-arbitrage conditions and we specify the
experimental design parameters that are needed for adequate performance. A
novel component is the use of the Dupire formula to enforce bounds on the local
volatility associated with option prices, during the network fitting. Our
methodology is benchmarked numerically on real datasets of DAX vanilla options.
