Forecasting implied volatility surface with generative diffusion models

We introduce a conditional Denoising Diffusion Probabilistic Model (DDPM) for generating arbitrage-free implied volatility (IV) surfaces, offering a more stable and accurate alternative to existing GAN-based approaches. To capture the path-dependent nature of volatility dynamics, our model is conditioned on a rich set of market variables, including exponential weighted moving averages (EWMAs) of historical surfaces, returns and squared returns of underlying asset, and scalar risk indicators like VIX. Empirical results demonstrate our model significantly outperforms leading GAN-based models in capturing the stylized facts of IV dynamics. A key challenge is that historical data often contains small arbitrage opportunities in the earlier dataset for training, which conflicts with the goal of generating arbitrage-free surfaces. We address this by incorporating a standard arbitrage penalty into the loss function, but apply it using a novel, parameter-free weighting scheme based on the signal-to-noise ratio (SNR) that dynamically adjusts the penalty's strength across the diffusion process. We also show a formal analysis of this trade-off and provide a proof of convergence showing that the penalty introduces a small, controllable bias that steers the model toward the manifold of arbitrage-free surfaces while ensuring the generated distribution remains close to the real-world data.