A Geometry-Aware Residual Correction of Hagan's SABR Implied Volatility Formula
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.
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BibTeX
@misc{reghai2026geometryaware,
title = {{A Geometry-Aware Residual Correction of Hagan's SABR Implied Volatility Formula}},
author = {Adil Reghai and Lama Tarsissi and Gérard Biau and Alex Lipton},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.06604},
eprint = {2605.06604},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.06604},
}
RIS
TY - UNPB
TI - A Geometry-Aware Residual Correction of Hagan's SABR Implied Volatility Formula
AU - Adil Reghai
AU - Lama Tarsissi
AU - Gérard Biau
AU - Alex Lipton
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2605.06604
UR - https://arxiv.org/abs/2605.06604
AB - 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.
ER -