This paper presents a novel neural network model for predicting the implied volatility surface (IVS) that explicitly incorporates prior financial domain knowledge. The authors design a custom activation function (smile function) that mimics the volatility smile pattern, applied to hidden nodes processing the log forward moneyness. Additionally, financial constraints such as no-arbitrage conditions (monotonicity, butterfly arbitrage), boundary conditions, and asymptotic slope are embedded as penalty terms in the loss function. The architecture consists of multiple single networks (each with smile and sigmoid activations) whose outputs are combined via a weighting network, allowing flexible modeling of the IVS.

Using 20 years of S&P 500 index options data (1996-2016), the proposed model (Multi) is compared against the state-of-the-art SSVI model and several neural network baselines (Single, Vanilla). Multi achieves the lowest mean absolute percentage error (MAPE) for both implied volatility (1.74% training, 3.34% test) and option prices (5.97% training, 10.64% test). Importantly, the model empirically satisfies the embedded financial constraints, ensuring consistency with established financial theory. This work demonstrates a successful framework for integrating domain knowledge into neural network design and training, enhancing both predictive performance and interpretability in quantitative finance.