This paper develops two neural-network-based frameworks for option pricing that incorporate financial option pricing PDEs while accommodating deviations from strict riskneutral valuation. The first approach, termed QINN, extends physics-informed neural networks (PINNs) by introducing a regularization parameter α that balances empirical data fitting with PDE-consistency. This enables QINN both to approximate PDE solutions directly and to infer latent model parameters, offering an alternative to conventional calibration techniques. The second approach, QINN 2 , removes the need for a prespecified model by embedding the volatility parameter of the Black–Scholes PDE into a separate neural network. This model-free formulation flexibly adapts to option data generated from different local and stochastic volatility models within a single framework. Numerical experiments across Black–Scholes, CEV, Heston, and 3/2 models demonstrate that QINN 2 matches or surpasses the accuracy of model-based QINN, especially for more complex dynamics. Together, these results highlight QINN and QINN 2 as practical and robust neural approaches to option pricing, bridging data-driven learning with financial model structure.
Kee et al. (Wed,) studied this question.