Before revision: This paper presents a framework of developing neural networks to predict implied volatility surfaces. It can incorporate the related properties from existing mathematical models and empirical findings, including no static arbitrage, limiting boundaries , asymptotic slopeand volatility smile. These properties are also satisfied empirically in our experiments with the option data on the S&P 500 index over 20 years.
After revision: In this paper we develop a novel neural network model for predicting implied volatility surface. Prior financial domain knowledge is taken into account. A new activation function that incorporates volatility smile is proposed, which is used for the hidden nodes that process the underlying asset price. In addition, financial conditions, such as the absence of arbitrage, the boundaries and the asymptotic slope, are embedded into the loss function. This is one of the very first studies which discuss a methodological framework that incorporates prior financial domain knowledge into neural network architecture design and model training. The proposed model outperforms the benchmarked models with the option data on the S&P 500 index over 20 years.