A Risk-Neutral Neural Operator for Arbitrage-Free SPX-VIX Term Structures
Jian'an Zhang
TL;DR
ARBITER reframes risk‑neutral pricing of SPX–VIX term structures as a structured neural operator tuned to arbitrage‑free and martingale constraints. It combines a risk‑neutral Green operator with a Lipschitz‑controlled Q‑Align layer and a convex–monotone SPX–VIX decoder, trained via saddle‑point optimization with explicit stopping criteria. The framework delivers dimensionless metrics (NAS/CNAS/NI/DualGap/Stability/SW/GenGap) with HAC‑CI and Holm‑Bonferroni corrections, demonstrating strong out‑of‑sample performance and robustness under ablations. Theoretical results guarantee approximation, identifiability, and convergence within a safety‑certified loop, while empirical analyses on a high‑fidelity synthetic SPX–VIX generator illustrate coherent pricing, stable calibration, and interpretable diagnostics. The work provides a reproducible blueprint for safety‑first operator learning in finance, with potential extensions to multi‑market coupling and more realistic market frictions.
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.