  We propose a fully data-driven approach to calibrate local stochastic
volatility (LSV) models, circumventing in particular the ad hoc interpolation
of the volatility surface. To achieve this, we parametrize the leverage
function by a family of feed-forward neural networks and learn their parameters
directly from the available market option prices. This should be seen in the
context of neural SDEs and (causal) generative adversarial networks: we
generate volatility surfaces by specific neural SDEs, whose quality is assessed
by quantifying, possibly in an adversarial manner, distances to market prices.
The minimization of the calibration functional relies strongly on a variance
reduction technique based on hedging and deep hedging, which is interesting in
its own right: it allows the calculation of model prices and model implied
volatilities in an accurate way using only small sets of sample paths. For
numerical illustration we implement a SABR-type LSV model and conduct a
thorough statistical performance analysis on many samples of implied volatility
smiles, showing the accuracy and stability of the method.
