Derivatives of feed-forward neural networks and their application in real-time market risk management

Market risk management of financial derivatives requires the efficient calculation of their price sensitivities with respect to changes in market factors. This paper shows how a deep feed-forward neural network which has been trained for pricing derivative instruments can be efficiently used to calculate these sensitivities as well. The proposed method is a fast and easily implementable alternative approach to automatic differentiation, and it simultaneously calculates all the first- and second-order derivatives of a multilayer feed-forward neural network with respect to its input features. The paper quantifies the performance improvement of the proposed method over a recent, publicly available implementation of automatic differentiation for a wide range of network sizes. The number of input parameters in these networks corresponds to those of commonly used financial models with stochastic volatility. The numerical accuracy of the proposed sensitivity calculations is demonstrated with a case study, calculating price sensitivities of European options under stochastic volatility. While the paper focuses on financial applications, the results presented herein are applicable to all deep feed-forward neural networks with sufficiently smooth activation functions.