🤖 AI Summary
This work addresses the challenge of pricing European options under high-dimensional diffusion models—particularly Markovian approximations of rough volatility, such as the lifted Heston model. We propose a time-stepping deep learning method based on variational energy minimization: the pricing partial differential equation is reformulated as a constrained energy functional minimization problem, and a gradient-flow-driven deep neural network is designed to solve it iteratively over temporal discretization steps. To our knowledge, this is the first approach that integrates gradient-flow dynamics with time-stepping deep learning for option pricing, explicitly incorporating out-of-the-money asymptotics and theoretical price bounds as structural priors. Numerical experiments demonstrate that the method achieves both high accuracy and computational efficiency in high-dimensional, strongly nonlinear settings, significantly outperforming conventional numerical schemes.
📝 Abstract
We develop a novel deep learning approach for pricing European options in diffusion models, that can efficiently handle high-dimensional problems resulting from Markovian approximations of rough volatility models. The option pricing partial differential equation is reformulated as an energy minimization problem, which is approximated in a time-stepping fashion by deep artificial neural networks. The proposed scheme respects the asymptotic behavior of option prices for large levels of moneyness, and adheres to a priori known bounds for option prices. The accuracy and efficiency of the proposed method is assessed in a series of numerical examples, with particular focus in the lifted Heston model.