Machine learning for option pricing: an empirical investigation of network architectures

The majority of articles in this literature considers a (plain) feed forward neural network architecture in order to connect the neurons used for learning the function mapping inputs to outputs. In this article, motivated by methods in image classification and recent advances in machine learning methods for PDEs, we investigate empirically whether and how the choice of network architecture affects the accuracy and training time of a machine learning algorithm. We find that for option pricing problems, where we focus on the Black-Scholes and the Heston model, the generalized highway network architecture outperforms all other variants, when considering the mean squared error and the training time as criteria. Moreover, for the computation of the implied volatility, after a necessary transformation, a variant of the DGM architecture outperforms all other variants, when considering again the mean squared error and the training time as criteria. Machine learning has taken the field of mathematical finance by a storm, and there are numerous applications of machine learning in finance by now. Concrete applications include, for example, the computation of option prices and implied volatilities as well as the calibration of financial models, see e.g. Buehler, Gonon, Teichmann, and Wood [10], portfolio selection and optimization, see e.g. A comprehensive overview of applications of machine learning in mathematical finance appears in the recent volume of Capponi and Lehalle [11], while an exhaustive overview focusing on pricing and hedging appears in Ruf and Wang [35]. We are interested in the computation of option prices and implied volatilities using machine learning methods, and thus, implicitly, in model calibration as well. More specifically, we consider the supervised learning problem of learning the price of an option or the implied volatility given appropriate input data (model parameters) and corresponding output data (option prices or implied volatilities). The majority of articles in this literature, see e.g. Cuchiero et al. [12], Horvath et al. [28], Liu et al. [30], consider a (plain) feed forward neural network architecture in order to connect the neurons used for learning the function mapping inputs to outputs. In this article, motivated by methods in image classification, see e.g. Option pricing, implied volatility, supervised learning, residual networks, highway networks, DGM networks. AP gratefully acknowledges the financial support from the Hellenic Foundation for Research and Innovation Grant No. HFRI-FM17-2152. JPH gratefully acknowledges the hospitality at the Financial Engineering & Mathematical Optimization Lab of the NTUA where this project was initiated. More specifically, next to the classical feed forward neural network or multilayer perceptron (MLP) architecture, we consider residual neural networks, highway networks and generalized highway networks.