  Albeit of crucial interest for both financial practitioners and researchers,
market-implied volatility data of European swaptions often exhibit large
portions of missing quotes due to illiquidity of the various underlying
swaption instruments. In this case, standard stochastic interpolation tools
like the common SABR model often cannot be calibrated to observed implied
volatility smiles, due to data being only available for the at-the-money quote
of the respective underlying swaption. Here, we propose to infer the geometry
of the full unknown implied volatility cube by learning stochastic latent
representations of implied volatility cubes via variational autoencoders,
enabling inference about the missing volatility data conditional on the
observed data by an approximate Gibbs sampling approach. Imputed estimates of
missing quotes can afterwards be used to fit a standard stochastic volatility
model. Since training data for the employed variational autoencoder model is
usually sparsely available, we test the robustness of the approach for a model
trained on synthetic data on real market quotes and we show that SABR
interpolated volatilites calibrated to reconstructed volatility cubes with
artificially imputed missing values differ by not much more than two basis
points compared to SABR fits calibrated to the complete cube. Moreover, we show
how the imputation can be used to successfully set up delta-neutral portfolios
for hedging purposes.
