Fast Calibration using Complex-Step Sobolev Training

We present a new fast calibration technique where we propose to train neural networks to directly perform the orthogonal projection of simulated payoffs of the calibration instrument with randomized model parameters and we enrich the learning task by including path-wise sensitivities of the payoffs with respect to model and product parameters. We show that this particular instance of Sobolev training can be reformulated in a way that requires computing only (stochastic) directional derivatives and we provide a fast, memory-efficient and numerically stable approach to compute those using complex-step differentiation. Our experiment with a fixed-grid piece-wise linear local volatility example demonstrates that one can get competitive price approximations without having to train the neural network on Monte Carlo prices and that both data-set construction and training can be done in reasonable time while preserving a very general framework that can be applied to a broad range of pricing models. We provide a highly optimized C++ code based on libtorch which includes all the necessary extensions for AAD on holomorphic neural networks on: https://github.com/BouazzaSE/TorchCSD