用神经算子建模股指与波动率曲线，确保无套利且精度更高。
A Risk-Neutral Neural Operator for Arbitrage-Free SPX-VIX Term Structures
- 通过约束解码器学习联合波动率曲线，强制满足无套利条件。
- 在历史数据上优于傅里叶神经算子等模型，长周期预测更稳定。
- 适合金融衍生品定价与风险控制场景，可实现跨期限插值外推。
我们提出ARBITER，一种用于学习无套利条件下标普500（SPX）与波动率指数（VIX）联合期限结构的风险中性神经算子。ARBITER将市场状态映射为算子，输出隐含波动率与方差曲线，并强制执行静态套利约束（日历、垂直、蝶式）、Lipschitz界与单调性。模型结合算子学习与受限解码器，采用外梯度风格更新及投影训练。引入新型评估指标：NAS、CNAS、NI、Dual-Gap和稳定性率。在历史SPX与VIX数据上，性能超越傅里叶神经算子、DeepONet及状态空间序列模型。消融实验表明：耦合SPX与VIX分支可降低Dual-Gap并提升NI；Lipschitz投影提升校准稳定性；选择性状态更新改善长期泛化能力。提供可识别性与近似性理论结果，并给出跨期限与行权价的无套利插值与外推实用方法。
原文摘要 · Abstract (English)
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.
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