Paper: arXiv 2511.06451

Authors: Jian’an Zhang

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 8.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents dense advanced mathematics, including neural operators, spectral projections, and convex optimization theory, justifying a high math score. It also demonstrates strong empirical rigor with specific evaluation metrics (NAS, CNAS, NI, etc.), historical data experiments, ablation studies, and confidence intervals, making it highly backtest-ready.

Research Flowchart

  flowchart TD
  A["Research Goal: Develop a risk-neutral neural operator\nfor arbitrage-free SPX-VIX term structures"] --> B["Methodology: ARBITER Framework"]
  B --> C["Data: Historical SPX & VIX Options\n(States, Volatility, Variance)"]
  C --> D["Computational Process:\nOperator Learning + Constrained Decoders\n(Enforcing no-arbitrage constraints)"]
  D --> E["Training: Extragradient + Projection\n(Static arbitrage, Lipschitz, Monotonicity)"]
  E --> F["Key Findings/Outcomes:\n1. Superior metrics (NAS, CNAS, NI, Dual-Gap)\n2. Reduced arbitrage violations\n3. Improved long-horizon generalization\n4. Practical recipes for interpolation/extrapolation"]

Cited References (96)

  1. Fourier Neural Operator for Parametric Partial Differential Equations (2021)
  2. Learning Nonlinear Operators via DeepONet Based on the Universal Approximation Theorem of Operators (2021)
  3. Spectral Normalization for Generative Adversarial Networks (2018)
  4. Sorting Out Lipschitz Function Approximation (2019)
  5. Input Convex Neural Networks (2017)
  6. Deep Lattice Networks and Partial Monotonic Functions (2017)
  7. A Tight and Unified Analysis of Extragradient for a Whole Spectrum of Variational Inequalities (2020)
  8. From Extra-Gradient to Coordinate Extra-Gradient Methods for Variational Inequalities (2022)
and 88 more
  1. Resurrecting Recurrent Neural Networks for Long Sequences (2023)
  2. A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix (1987)
  3. A Simple Sequentially Rejective Multiple Test Procedure (1979)
  4. Regularisation of Neural Networks by Enforcing Lipschitz Continuity (2021)
  5. Arbitrage-Free Construction of Implied Volatility Surfaces (2019)
  6. Arbitrage-Free Volatility Surfaces (2021)
  7. Lookahead Optimizer: k Steps Forward, 1 Step Back (2019)
  8. Mamba: Linear-Time Sequence Modeling with Selective State Spaces (2023) — link
  9. Efficiently Modeling Long Sequences with Structured State Spaces (2022)
  10. S5: Scalable State Space Models (2023) — link
  11. Hyena Hierarchy: Towards Larger Context and Longer Sequences (2023)
  12. It is Raw! Audio and Beyond with SSMs for Sequence Modeling (2023) — link
  13. Neural Operator: Learning Maps Between Function Spaces (2023)
  14. Data-Efficient Deep Operator Learning via Differentially Enhanced DeepONet (2024)
  15. Deep Smoothing of the Implied Volatility Surface (2020)
  16. Arbitrage-free Neural-SDE Market Models (2021) — link
  17. Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention (2020)
  18. Rethinking Attention with Performers (2021)
  19. On the Stability of Selective State Space Models (2024) — link
  20. Lipschitz Neural Networks: A Survey (2023) — link
  21. Computational Optimal Transport: With Applications to Data Science (2019)
  22. Volatility Index Methodology – Cboe Volatility Index (VIX) (2025)
  23. Neural Networks for Option Pricing: A Survey (2020)
  24. Deep Neural Stochastic PDEs for Financial Modeling (2022)
  25. Arbitrage-Free Yield Curve Modeling with Neural Networks (2020)
  26. Machine Learning under the Risk-Neutral Measure (2023)
  27. Deep Hedging (2019)
  28. Arbitrage-Free SVI Volatility Surfaces (2014)
  29. Data-driven Option Pricing (2024) — link
  30. Is Mamba Effective for Time Series Forecasting? (2024) — link
  31. Robust Neural Operators via Lipschitz Regularization (2024)
  32. Arbitrage-Free Implied Volatility Surface Generation with Variational Autoencoders (2023)
  33. Neural Network Approaches to Implied Volatility Surfaces (2019) — link
  34. Risk-Neutral Pricing in Deep Learning Frameworks (2021) — link
  35. The M5 Competition: Results, Findings, and Conclusions (2022)
  36. Convex Neural Calibration of Option Surfaces (2022) — link
  37. State Space Models for Deep Sequence Modeling: A Review — → 2401.00001
  38. Convex Optimization Theory (2015)
  39. The Cboe Volatility Index – VIX (2019)
  40. Learning to Simulate Complex Physics with Graph Networks (2020)
  41. Rademacher and Gaussian Complexities: Risk Bounds and Structural Results (2002)
  42. Prox-Method with Rate of Convergence O(1/t) for Variational Inequalities with Lipschitz Continuous Monotone Operators (2004)
  43. Dual Extrapolation and Its Applications to Solving Variational Inequalities and Related Problems (2007)
  44. Variational Inequality Perspective on Generative Adversarial Networks (2019)
  45. Optimistic Mirror Descent in Saddle-Point Problems: Going the Extra (Gradient) Mile (2019)
  46. The Singular Values of Convolutional Layers (2019)
  47. Pricing Under Rough Volatility (2016)
  48. Functional It ^o Calculus and Volatility Modelling (2019)
  49. A Stochastic Volatility Model with Regime Switching and Fast Mean-Reversion (2019)
  50. Fourier Neural Operator for Parametric Partial Differential Equations (2021)
  51. Learning Nonlinear Operators via DeepONet Based on the Universal Approximation Theorem of Operators (2021)
  52. Neural Operator: Learning Maps between Function Spaces with Applications to PDEs (2023)
  53. Message Passing Neural PDE Solvers (2022)
  54. U-NO: U-shaped Neural Operators (2022)
  55. Wavelet Neural Operator for Solving Parametric Partial Differential Equations (2022)
  56. Physics-Informed Neural Operator for Learning PDEs (2021)
  57. Physics-Informed Neural Operators: A Review (2023)
  58. Error Estimates for DeepONets: A Deep Learning Framework in Infinite Dimensions (2022)
  59. Stability and Generalization of Operator Learning with Applications to Scientific Machine Learning (2023)
  60. Learning to Simulate Complex Physics with Graph Networks (2020)
  61. Efficiently Modeling Long Sequences with Structured State Spaces (2022)
  62. Simplifying and Stabilizing S4 for Efficient Sequence Modeling (2023) — link
  63. S5: Scalable State Space Models (2023) — link
  64. Hyena Hierarchy: Towards Larger Contexts and Longer Sequences (2023)
  65. Retentive Networks: A Successor to Transformers (2024)
  66. Mamba: Linear-Time Sequence Modeling with Selective State Spaces (2024)
  67. VMamba: Visual State Space Modeling (2024)
  68. Mamba in Speech: Towards an Alternative to Self-Attention (2024) — link
  69. Arbitrage-Free Construction of Implied Volatility Surfaces (2019)
  70. Arbitrage-Free Volatility Surfaces (2021)
  71. Deep Calibration of Rough Stochastic Volatility Models (2021)
  72. OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport (2021)
  73. Flow Matching for Generative Modeling (2023)
  74. Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow (2023)
  75. Neural Controlled Differential Equations for Irregular Time Series (2020)
  76. Scalable Gradients and Variational Inference for Stochastic Differential Equations (2020)
  77. Deep Learning in Mathematical Finance (2022)
  78. Martingale Optimal Transport and Robust Finance: A Survey (2020)
  79. Recent Advances in Martingale Optimal Transport (2022)
  80. Arbitrage-Free Volatility Surfaces: Parametric Representations Revisited (2023)
  81. Signature SDE Models in Mathematical Finance (2020)
  82. Machine Learning for Local Volatility Calibration (2020)
  83. Risk-Neutral Learning and Arbitrage Constraints in Deep Option Pricing (2022)
  84. Arbitrage-Free SVI Volatility Surfaces (2020)
  85. Rough Volatility: A Practitioner’s Guide (2023)
  86. Learning Term Structures: From HJM to Deep Generative Models (2021)
  87. Cross-Validation Strategies for Data with Temporal, Spatial, Hierarchical, or Phylogenetic Structure (2017)
  88. Exploring Generalization in Deep Learning (2017)

1 of 96 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.