Paper: arXiv 2605.06604

Authors: Adil Reghai, Lama Tarsissi, Gérard Biau, Alex Lipton

Abstract

This paper proposes a hybrid methodology to improve the approximation of SABR (Stochastic Alpha Beta Rho) implied volatility by combining analytical structure with machine learning. The approach augments the neural-network input representation with geometric features derived from the stochastic differential equations of the SABR model. Unlike approaches that fully replace analytical formulas with black-box models, the proposed framework preserves the analytical backbone of the model. The hybridization operates along two complementary dimensions. First, geometry-aware variables reflecting intrinsic properties of the SABR dynamics are used as structured inputs to the network. Second, the neural network is trained to learn the residual error relative to Hagan’s closed-form approximation rather than implied volatility directly. The resulting model acts as a structured residual correction to the analytical formula, retaining interpretability while capturing higher-order effects that are not included in the asymptotic expansion. Numerical experiments conducted over realistic parameter domains, as well as stressed environments, show that the method improves accuracy and robustness compared with both analytical approximations and standard neural-network approaches. Because the correction remains lightweight and structurally consistent with the underlying model, the framework is well suited for real-time pricing and calibration in practical trading environments.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper combines sophisticated differential geometry and stochastic analysis (SABR dynamics, Riemannian metrics) with a high-fidelity, variance-reduced Monte Carlo data pipeline for training and empirical validation. This integration of rigorous mathematical structure with a production-ready residual learning framework places it squarely in the Holy Grail category.

Research Flowchart

  flowchart TD
    A[Research Goal: Improve SABR Implied Volatility Accuracy] --> B[Data: SABR Model Parameters & Stochastic Dynamics]
    B --> C[Feature Engineering: Geometry-Aware Input Representation]
    C --> D[Hybrid Modeling: NN Learning Residual of Hagan's Formula]
    D --> E[Computational Process: Training & Residual Correction]
    E --> F[Outcomes: High Accuracy, Robustness, & Real-time Efficiency]

Cited References (27)

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  4. Managing Vol Surfaces (2018) — link
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  7. Neural Networks with Asymptotics Control (2020)
  8. The CV makes the difference – C ontrol variates for neural networks (2020)
and 19 more
  1. Artificial neural network for option pricing with and without asymptotic correction (2020)
  2. SABR equipped with AI wings (2023) — link
  3. Extensive networks would eliminate the demand for pricing formulas (2022) — link
  4. No-Arbitrage Deep Calibration for Volatility Smile and Skewness — → 2310.16703
  5. Deep learning of transition probability densities for stochastic asset models with applications in option pricing (2025) — link
  6. Learning the Exact SABR Model — → 2510.10343
  7. The study on option pricing based on W iener– I t ^o chaos expansion and generative adversarial networks (2025)
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  9. Exact simulation of the SABR model (2017) — link
  10. Modelling the volatility smile in negative-rate environments: E mpirical analysis of SABR variants (2023)
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  12. An efficient calibration of stochastic volatility models using neural networks (2019)
  13. Incorporating prior financial domain knowledge into neural networks for implied volatility surface prediction (2019)
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  18. Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing (2009)
  19. Financial Models in Production (2020)

2 of 27 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.