Learning the Exact SABR Model
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.