  We explore the abilities of two machine learning approaches for no-arbitrage
interpolation of European vanilla option prices, which jointly yield the
corresponding local volatility surface: a finite dimensional Gaussian process
(GP) regression approach under no-arbitrage constraints based on prices, and a
neural net (NN) approach with penalization of arbitrages based on implied
volatilities. We demonstrate the performance of these approaches relative to
the SSVI industry standard. The GP approach is proven arbitrage-free, whereas
arbitrages are only penalized under the SSVI and NN approaches. The GP approach
obtains the best out-of-sample calibration error and provides uncertainty
quantification.The NN approach yields a smoother local volatility and a better
backtesting performance, as its training criterion incorporates a local
volatility regularization term.
