A Risk-Neutral Neural Operator for Arbitrage-Free SPX-VIX Term Structures

TL;DR

ARBITER is a neural operator that learns arbitrage-free joint SPX-VIX term structures, ensuring no-arbitrage constraints and improving calibration stability and accuracy over existing models.

Contribution

The paper introduces ARBITER, a novel risk-neutral neural operator that enforces no-arbitrage constraints in joint SPX-VIX term structure modeling, with new evaluation metrics and practical calibration methods.

Findings

ARBITER outperforms Fourier Neural Operator, DeepONet, and state-space models on historical data.

Tying SPX and VIX legs reduces Dual-Gap and enhances NI.

Lipschitz projection stabilizes calibration and improves long-term generalization.

Abstract

We propose ARBITER, a risk-neutral neural operator for learning joint SPX-VIX term structures under no-arbitrage constraints. ARBITER maps market states to an operator that outputs implied volatility and variance curves while enforcing static arbitrage (calendar, vertical, butterfly), Lipschitz bounds, and monotonicity. The model couples operator learning with constrained decoders and is trained with extragradient-style updates plus projection. We introduce evaluation metrics for derivatives term structures (NAS, CNAS, NI, Dual-Gap, Stability Rate) and show gains over Fourier Neural Operator, DeepONet, and state-space sequence models on historical SPX and VIX data. Ablation studies indicate that tying the SPX and VIX legs reduces Dual-Gap and improves NI, Lipschitz projection stabilizes calibration, and selective state updates improve long-horizon generalization. We provide identifiability and approximation results and describe practical recipes for arbitrage-free interpolation and extrapolation across maturities and strikes.