A two-step framework for arbitrage-free prediction of the implied volatility surface

In this study, we propose a two-step framework to predict the implied volatility surface (IVS) in a manner that excludes static arbitrage. First, we select features to represent the surface and predict them. Second, we use the predicted features to construct the IVS using a deep neural network (DNN) model by incorporating constraints that can prevent static arbitrage. We consider three methods to extract features from the implied volatility data: principal component analysis, variational autoencoder, and sampling the surface. We predict these features using the long short-term memory model. Additionally, we use a long time series of implied volatility data for S&P500 index options to train our models. We find that two feature construction methods (i.e. sampling the surface and variational autoencoders combined with DNN for surface construction) are the best performers in the out-of-sample prediction. Furthermore, both of them substantially outperform a popular regression model. We also find that the DNN model for surface construction not only removes static arbitrage but also significantly reduces the prediction error compared with a standard interpolation method.