Arbitrage-Free Implied Volatility Surface Reconstruction via Additive Martingales

This work focuses on reconstructing a continuous implied volatility (IV) surface from a small number of option quotes, a fundamental task for option pricing and risk management. We propose Additive Martingale (AddMart), a few-shot IV surface reconstruction framework that satisfies static no-arbitrage conditions by construction. Rather than predicting IVs directly, AddMart maps sparse quotes to non-negative increment densities of an exponential additive martingale, from which valid risk-neutral marginals and IVs are recovered. By incorporating the no-arbitrage structure directly into its architecture, rather than enforcing it through regularization in the learning objective, AddMart produces IV surfaces that satisfy static no-arbitrage conditions throughout the domain. Extensive experiments on ten real-option datasets demonstrate that AddMart achieves competitive reconstruction accuracy with no material static arbitrage violations, while directly enabling the estimation of VaR, CVaR, and crash probabilities.