This paper presents a framework of developing neural networks for predicting implied volatility surfaces. Conventional financial conditions and empirical evidence related to the implied volatility are incorporated into the neural network architecture design and model training including no static arbitrage,  boundaries, asymptotic slope and volatility smile. They are also satisfied empirically by the option data on the S&P 500 index over twenty years. The developed neural network model and its simplified variations outperform the widely used surface stochastic volatility inspired (SSVI) model  on the mean average percentage error in both in-sample and out-of-sample datasets. This study has two main methodological contributions. First,  an accurate deep learning  prediction model is developed and tailored to implied volatility surfaces . Second, a framework, which seamlessly combines data-driven models with financial theories, can be extended and applied to solve other related business problems .
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 slope and volatility smile. These properties are also satisfied empirically in our experiments with the option data on the S&P 500 index over 20 years. The developed neural network model outperforms the widely used surface stochastic volatility inspired (SSVI) model and other benchmarked neural network models on the mean average percentage error in both in-sample and out-of-sample datasets. This study has two major contributions. First, it contributes to the recent use of machine learning in finance, and an accurate deep learning implied volatility surface prediction model is obtained . Second, it provides the methodological guidance on how to seamlessly combine data-driven models with domain knowledge in the development of machine learning applications .
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financial theories, can be extended and applied to solve other related business problems
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