Additive martingales reconstruct accurate option volatility surfaces without arbitrage
Arbitrage-Free Implied Volatility Surface Reconstruction via Additive Martingales
Machine Learning
Summary
Option prices depend on a surface called the implied volatility surface, which is needed for pricing and managing risks but is hard to build accurately from limited data. The authors propose AddMart, a method that uses a mathematical tool called additive martingales to create this surface in a way that automatically avoids impossible pricing scenarios called arbitrage. Instead of directly predicting volatility, AddMart predicts certain underlying probabilities and derives valid prices from them. Their method works well on real market data and helps estimate important risk measures like value-at-risk.
What this means in practice
- •For financial risk managers: Generate consistent implied volatility surfaces from sparse option quotes to improve risk estimation accuracy.
- •For quantitative traders: Produce arbitrage-free volatility surfaces that enable reliable assessment of crash probabilities and tail risks.
Authors
Haochen Luo, Sifan Lin, Xiaolong Luo, An Binh Minh, Yuan Zhang, Chen Liu
Abstract
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.