A time-stepping deep gradient flow method for option pricing in (rough) diffusion 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. Stochastic volatility models have been popular in the mathematical finance literature because they allow to accurately model and reproduce the shape of implied volatility smiles for a single maturity. They require though certain modifications, such as making the parameters time-or maturity-dependent, in order to reproduce a whole volatility surface; see e.g. the comprehensive books by Gatheral [25] or Bergomi [15]. The class of rough volatility models, in which the volatility process is driven by a fractional Brownian motion, offers an attractive alternative to classical volatility models, since they allow to reproduce many stylized facts of asset and option prices with only a few (constant) parameters; see e.g. the seminal articles by Gatheral, Jaisson, and Rosenbaum [27] and Bayer, Friz, and Gatheral [9], and the recent volume by Bayer, Friz, Fukasawa, Gatheral, Jacquier, and Rosenbaum [13].