SABR Model

Learning the Exact SABR Model ArXiv ID: 2510.10343 “View on arXiv” Authors: Giorgia Rensi, Pietro Rossi, Marco Bianchetti Abstract The SABR model is a cornerstone of interest rate volatility modeling, but its practical application relies heavily on the analytical approximation by Hagan et al., whose accuracy deteriorates for high volatility, long maturities, and out-of-the-money options, admitting arbitrage. While machine learning approaches have been proposed to overcome these limitations, they have often been limited by simplified SABR dynamics or a lack of systematic validation against the full spectrum of market conditions. We develop a novel SABR DNN, a specialized Artificial Deep Neural Network (DNN) architecture that learns the true SABR stochastic dynamics using an unprecedented large training dataset (more than 200 million points) of interest rate Cap/Floor volatility surfaces, including very long maturities (30Y) and extreme strikes consistently with market quotations. Our dataset is obtained via high-precision unbiased Monte Carlo simulation of a special scaled shifted-SABR stochastic dynamics, which allows dimensional reduction without any loss of generality. Our SABR DNN provides arbitrage-free calibration of real market volatility surfaces and Cap/Floor prices for any maturity and strike with negligible computational effort and without retraining across business dates. Our results fully address the gaps in the previous machine learning SABR literature in a systematic and self-consistent way, and can be extended to cover any interest rate European options in different rate tenors and currencies, thus establishing a comprehensive functional SABR framework that can be adopted for daily trading and risk management activities. ...

Meta-Learning Neural Process for Implied Volatility Surfaces with SABR-induced Priors ArXiv ID: 2509.11928 “View on arXiv” Authors: Jirong Zhuang, Xuan Wu Abstract We treat implied volatility surface (IVS) reconstruction as a learning problem guided by two principles. First, we adopt a meta-learning view that trains across trading days to learn a procedure that maps sparse option quotes to a full IVS via conditional prediction, avoiding per-day calibration at test time. Second, we impose a structural prior via transfer learning: pre-train on SABR-generated dataset to encode geometric prior, then fine-tune on historical market dataset to align with empirical patterns. We implement both principles in a single attention-based Neural Process (Volatility Neural Process, VolNP) that produces a complete IVS from a sparse context set in one forward pass. On SPX options, the VolNP outperforms SABR, SSVI, and Gaussian process. Relative to an ablation trained only on market data, the SABR-induced prior reduces RMSE by about 40% and suppresses large errors, with pronounced gains at long maturities where quotes are sparse. The resulting model is fast (single pass), stable (no daily recalibration), and practical for deployment at scale. ...

SABR-Informed Multitask Gaussian Process: A Synthetic-to-Real Framework for Implied Volatility Surface Construction ArXiv ID: 2506.22888 “View on arXiv” Authors: Jirong Zhuang, Xuan Wu Abstract Constructing the Implied Volatility Surface (IVS) is a challenging task in quantitative finance due to the complexity of real markets and the sparsity of market data. Structural models like Stochastic Alpha Beta Rho (SABR) model offer interpretability and theoretical consistency but lack flexibility, while purely data-driven methods such as Gaussian Process regression can struggle with sparse data. We introduce SABR-Informed Multi-Task Gaussian Process (SABR-MTGP), treating IVS construction as a multi-task learning problem. Our method uses a dense synthetic dataset from a calibrated SABR model as a source task to inform the construction based on sparse market data (the target task). The MTGP framework captures task correlation and transfers structural information adaptively, improving predictions particularly in data-scarce regions. Experiments using Heston-generated ground truth data under various market conditions show that SABR-MTGP outperforms both standard Gaussian process regression and SABR across different maturities. Furthermore, an application to real SPX market data demonstrates the method’s practical applicability and its ability to produce stable and realistic surfaces. This confirms our method balances structural guidance from SABR with the flexibility needed for market data. ...

Integrating the implied regularity into implied volatility models: A study on free arbitrage model ArXiv ID: 2502.07518 “View on arXiv” Authors: Unknown Abstract Implied volatility IV is a key metric in financial markets, reflecting market expectations of future price fluctuations. Research has explored IV’s relationship with moneyness, focusing on its connection to the implied Hurst exponent H. Our study reveals that H approaches 1/2 when moneyness equals 1, marking a critical point in market efficiency expectations. We developed an IV model that integrates H to capture these dynamics more effectively. This model considers the interaction between H and the underlying-to-strike price ratio S/K, crucial for capturing IV variations based on moneyness. Using Optuna optimization across multiple indexes, the model outperformed SABR and fSABR in accuracy. This approach provides a more detailed representation of market expectations and IV-H dynamics, improving options pricing and volatility forecasting while enhancing theoretical and pratcical financial analysis. ...

Efficient and accurate simulation of the stochastic-alpha-beta-rho model ArXiv ID: 2408.01898 “View on arXiv” Authors: Unknown Abstract We propose an efficient, accurate and reliable simulation scheme for the stochastic-alpha-beta-rho (SABR) model. The two challenges of the SABR simulation lie in sampling (i) integrated variance conditional on terminal volatility and (ii) terminal forward price conditional on terminal volatility and integrated variance. For the first sampling procedure, we sample the conditional integrated variance using the moment-matched shifted lognormal approximation. For the second sampling procedure, we approximate the conditional terminal forward price as a constant-elasticity-of-variance (CEV) distribution. Our CEV approximation preserves the martingale condition and precludes arbitrage, which is a key advantage over Islah’s approximation used in most SABR simulation schemes in the literature. We then adopt the exact sampling method of the CEV distribution based on the shifted-Poisson mixture Gamma random variable. Our enhanced procedures avoid the tedious Laplace inversion algorithm for sampling integrated variance and non-efficient inverse transform sampling of the forward price in some of the earlier simulation schemes. Numerical results demonstrate our simulation scheme to be highly efficient, accurate, and reliable. ...

Reinforcement Learning and Deep Stochastic Optimal Control for Final Quadratic Hedging ArXiv ID: 2401.08600 “View on arXiv” Authors: Unknown Abstract We consider two data driven approaches, Reinforcement Learning (RL) and Deep Trajectory-based Stochastic Optimal Control (DTSOC) for hedging a European call option without and with transaction cost according to a quadratic hedging P&L objective at maturity (“variance-optimal hedging” or “final quadratic hedging”). We study the performance of the two approaches under various market environments (modeled via the Black-Scholes and/or the log-normal SABR model) to understand their advantages and limitations. Without transaction costs and in the Black-Scholes model, both approaches match the performance of the variance-optimal Delta hedge. In the log-normal SABR model without transaction costs, they match the performance of the variance-optimal Barlett’s Delta hedge. Agents trained on Black-Scholes trajectories with matching initial volatility but used on SABR trajectories match the performance of Bartlett’s Delta hedge in average cost, but show substantially wider variance. To apply RL approaches to these problems, P&L at maturity is written as sum of step-wise contributions and variants of RL algorithms are implemented and used that minimize expectation of second moments of such sums. ...

No-Arbitrage Deep Calibration for Volatility Smile and Skewness ArXiv ID: 2310.16703 “View on arXiv” Authors: Unknown Abstract Volatility smile and skewness are two key properties of option prices that are represented by the implied volatility (IV) surface. However, IV surface calibration through nonlinear interpolation is a complex problem due to several factors, including limited input data, low liquidity, and noise. Additionally, the calibrated surface must obey the fundamental financial principle of the absence of arbitrage, which can be modeled by various differential inequalities over the partial derivatives of the option price with respect to the expiration time and the strike price. To address these challenges, we have introduced a Derivative-Constrained Neural Network (DCNN), which is an enhancement of a multilayer perceptron (MLP) that incorporates derivatives in the objective function. DCNN allows us to generate a smooth surface and incorporate the no-arbitrage condition thanks to the derivative terms in the loss function. In numerical experiments, we train the model using prices generated with the SABR model to produce smile and skewness parameters. We carry out different settings to examine the stability of the calibrated model under different conditions. The results show that DCNNs improve the interpolation of the implied volatility surface with smile and skewness by integrating the computation of the derivatives, which are necessary and sufficient no-arbitrage conditions. The developed algorithm also offers practitioners an effective tool for understanding expected market dynamics and managing risk associated with volatility smile and skewness. ...