Pricing financial derivatives such as options is a key challenge in financial mathematics. A common approach is to express the option price as the solution to a partial differential equation (PDE). As option pricing models become increasingly complicated, the resulting PDEs become high-dimensional, rendering classical numerical methods computationally infeasible. Recent advances suggest that deep learning may offer efficient alternatives for solving such problems.

This dissertation develops a novel deep learning approach for pricing options, which can efficiently handle high-dimensional problems. We call our method the Time Deep Gradient Flow (TDGF). The TDGF reformulates the PDE as a time-discretized variational problem, which is then approximated using deep neural networks. We focus on three option pricing models: the Black--Scholes model with constant volatility; the Heston model with stochastic volatility and the lifted Heston model, a Markovian approximation of rough volatility. We compare our method in these models with classical numerical methods such as Monte Carlo and Fourier methods and another deep learning method called the Deep Galerkin Method (DGM).

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.