  We propose a hybrid method for generating arbitrage-free implied volatility
(IV) surfaces consistent with historical data by combining model-free
Variational Autoencoders (VAEs) with continuous time stochastic differential
equation (SDE) driven models. We focus on two classes of SDE models: regime
switching models and L\'evy additive processes. By projecting historical
surfaces onto the space of SDE model parameters, we obtain a distribution on
the parameter subspace faithful to the data on which we then train a VAE.
Arbitrage-free IV surfaces are then generated by sampling from the posterior
distribution on the latent space, decoding to obtain SDE model parameters, and
finally mapping those parameters to IV surfaces. We further refine the VAE
model by including conditional features and demonstrate its superior generative
out-of-sample performance.
