  We propose a gradient-based deep learning framework to calibrate the Heston
option pricing model (Heston, 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 gradient-based 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.
