SHoP-Based Neural PDE Framework for Solving Black–Scholes Pricing Equations

Solving high-dimensional financial partial differential equations remains challenging due to the curse of dimensionality, costly higher-order derivatives, and limited interpretability. We propose SHoP-BS, an efficient and interpretable neural solver for Black–Scholes (BS) pricing PDEs built upon the SHoP framework. The method focuses on representative financial scenarios including European down-and-out barrier options and multiasset rainbow options derived from the BS equation family. SHoP-BS uses a matrix-based high-order derivative rule to obtain first- and second-order (including mixed) partials in one pass, greatly reducing computational overhead while stabilizing PDE-residual training and boundary/terminal enforcement. A local Taylor expansion yields an explicit and interpretable representation of the option value and the Greeks. Using synthetically generated data with analytical or high-quality Monte Carlo references, experiments show that SHoP-BS attains higher global accuracy, better satisfaction of boundary conditions, and smoother, more physically consistent Greeks than finite differences, and delivers more stable convergence and higher inference efficiency than autograd-based PINNs when multiple sensitivities are required. Overall, SHoP-BS provides a unified paradigm that balances efficiency, accuracy, and interpretability for BS-type pricing PDEs with higher-order derivatives.