This paper introduces a novel hybrid method for generating arbitrage-free implied volatility (IV) surfaces by combining Variational Autoencoders (VAEs) with stochastic differential equation (SDE) models. The authors focus on two classes of SDE models: regime-switching continuous-time Markov chains (CTMC) and Lévy additive processes, both of which are inherently arbitrage-free. The SDE models are calibrated to historical market data using a loss function that includes a 1-Wasserstein distance penalty to prevent overfitting, especially in sparse data settings like FX options. The resulting SDE parameters form a distribution that is then learned by a VAE, which allows for generating new parameter samples from the latent space. These samples are decoded and mapped to IV surfaces, ensuring the generated surfaces are both realistic and arbitrage-free.

The paper extends the VAE framework to a conditional VAE (CVAE) that incorporates market features (e.g., spot price, interest rates) as conditioning variables, improving generative performance. The method is tested on 1,900 days of FX data for three currency pairs (AUD-USD, EUR-USD, CAD-USD). Results show that the CVAE outperforms standard VAE and β-VAE variants in out-of-sample generation, producing IV surfaces that closely match historical patterns while maintaining no-arbitrage conditions. The approach is particularly valuable for generating synthetic training data for deep learning pricing models and for risk management applications where realistic, arbitrage-free surface generation is critical.