Deep neural networks for the valuation of equity and real-estate index American options under models with stochastic volatility

Purpose The primary objective of this research is to develop new algorithms in the framework of deep neural networks for the valuation of options under dynamics driven by stochastic volatility models. We aim to use the Heston model for equity options to demonstrate the accuracy of our approach. Design/methodology/approach Physics-informed neural networks (PINNs) are trained to minimize a loss function that includes terms from the partial differential equation residuals, initial condition and boundary conditions evaluated at selected points in the space-time domain. Speed and accuracy comparisons are carried out against single hidden-layer neural networks, called physics-informed extreme learning machines (PIELMs). American options are formulated as linear complementarity problems, and PINNs are applied in conjunction with penalty methods for the computation of the option prices. Findings For American options under the Heston model, PINNs yield accurate prices. Computed Greeks sensitivities are in close agreement with those reported for mesh-based methods. In contrast to mesh-based penalty methods for American options, PINNs work with smaller values of the penalty term. For the real estate index American option problem, numerical prices obtained using PINNs have comparable accuracies as those obtained by a high-order radial basis functions finite difference scheme. Practical implications There is a lack of reliable pricing models for pricing property derivatives. This work contributes to developing accurate neural network algorithms. Originality/value The development of informed neural networks and informed extreme learning machines for realistic two-factor real-world option contracts constitutes a novel approach in the field of financial derivatives.