Physics-Informed Neural Network approach to time-fractional Black–Scholes models: Pricing down-and-in Parisian options under rough volatility

This paper investigates the pricing of down-and-in Parisian options within a time-fractional rough volatility Black–Scholes framework. The proposed model incorporates both fractional dynamics and rough volatility effects, allowing for the simultaneous representation of long-range dependence, memory effects, and highly irregular volatility paths observed in financial markets. The fractional component introduces non-local temporal behaviour, capturing persistence and subdiffusive dynamics, while the rough volatility structure, driven by a Volterra-type process with Hurst parameter H<0.5, accounts for the observed anti-persistent and oscillatory nature of volatility. The resulting time-fractional rough volatility Black–Scholes partial differential equation (tf-RVBSPDE) extends classical models by embedding both path dependence and global memory into the pricing mechanism. In addition, the Parisian feature is formulated through an occupation-time constraint, ensuring that barrier activation depends on sustained excursions rather than instantaneous crossings, thereby enhancing stability in volatile market conditions. To address the analytical and computational challenges posed by the non-local and non-Markovian structure of the model, a Physics-Informed Neural Network (PINN) approach is employed. The Grünwald–Letnikov formulation of the fractional derivative is adopted to ensure compatibility with automatic differentiation and efficient numerical implementation. The PINN method approximates the solution by minimizing a loss function that enforces both the governing equation and boundary conditions. Numerical experiments based on S&P 500 data demonstrate that the proposed approach accurately captures the effects of rough volatility and memory, with improved performance in regimes where α<0.5. The results confirm that the model recovers classical solutions in limiting cases and provides a robust pricing approach for cases when tractable analytical solutions are non-existing. Overall, the approach establishes a unified connection between fractional stochastic dynamics, rough volatility modelling, and PINN based numerical methods for pricing complex path-dependent derivative contracts.