Published Paper

Inserted: 16 mar 2026
Last Updated: 16 mar 2026

Journal: Decisions Econ Finan
Year: 2025
Doi: 10.1007/s10203-025-00558-1
Links: article on journal website

Abstract:

We propose a gradient-based deep learning framework to calibrate the Heston option pricing model S.L. Heston (Rev. Financ. Stud. 6(2), 327-343 (1993)). Our neural network, henceforth deep differential network (DDN), learns both the Heston pricing formula for plain-vanilla options and the partial derivatives with respect to the model parameters. The price sensitivities estimated by the DDN are not subject to the numerical issues that can be encountered in computing the gradient of the Heston pricing function. Thus, our network is an excellent pricing engine for fast gradientbased calibrations. Extensive tests on selected equity markets show that the DDN significantly outperforms non-differential feedforward neural networks in terms of calibration accuracy. In addition, it dramatically reduces the computational time with respect to global optimizers that do not use gradient information.

Keywords: stochastic volatility, Machine learning, deep differential neural networks, model calibration, option pricing