This paper introduces a novel gated neural network architecture for pricing European call options that combines data-driven learning with economic rationality constraints. The model consists of two components: a set of 'expert' pricing networks that each learn a local pricing function, and a gating network that softly assigns each option to the appropriate expert based on moneyness and time-to-maturity. This divide-and-conquer approach allows the model to automatically adapt to different market regimes without manual categorization. Crucially, the network is designed to inherently satisfy six no-arbitrage conditions (C1-C6), including monotonicity in strike and maturity, convexity in strike, and appropriate boundary behavior. For conditions that cannot be enforced architecturally (C5, C6), the authors synthesize virtual training data points. For the multi-model, a learning-from-hints loss is added to ensure convexity (C2).

Empirically, the model is evaluated on a large dataset of S&P 500 index options spanning 20 years (1996-2016) with over 3 million contracts. It significantly outperforms both econometric models (Black-Scholes, Variance Gamma, Kou Jump) and previous neural network approaches (PSSF, Modular Neural Networks) in terms of MSE and MAPE. The multi-model achieves a test MAPE of 6.84% compared to 26.25% for PSSF and 22.42% for Black-Scholes. Additionally, the model produces a valid risk-neutral density function, which is a key requirement for risk management applications. The authors demonstrate that the density is non-negative and integrates to one, unlike prior neural methods. Overall, the work shows that embedding domain knowledge (economic axioms) into neural network design not only guarantees economically sensible predictions but also improves generalization and predictive accuracy.