  We propose a two-step framework for predicting the implied volatility surface
over time without static arbitrage. In the first step, we select features to
represent the surface and predict them over time. In the second step, we use
the predicted features to construct the implied volatility surface using a deep
neural network (DNN) model by incorporating constraints that prevent static
arbitrage. We consider three methods to extract features from the implied
volatility data: principal component analysis, variational autoencoder and
sampling the surface, and we predict these features using LSTM. Using a long
time series of implied volatility data for S\&P500 index options to train our
models, we find two feature construction methods, sampling the surface and
variational autoencoders combined with DNN for surface construction, are the
best performers in out-of-sample prediction. In particular, they outperform a
classical method substantially. Furthermore, the DNN model for surface
construction not only removes static arbitrage, but also significantly reduces
the prediction error compared with a standard interpolation method. Our
framework can also be used to simulate the dynamics of the implied volatility
surface without static arbitrage.
