A Two-Step Framework for Arbitrage-Free Prediction of the Implied Volatility Surface

TL;DR

This paper introduces a two-step deep learning framework for predicting the implied volatility surface over time that avoids static arbitrage and outperforms classical methods in accuracy.

Contribution

It presents a novel two-step approach combining feature extraction and deep neural networks to predict and construct arbitrage-free implied volatility surfaces.

Findings

Sampling and variational autoencoder features outperform classical methods

Deep neural network construction reduces prediction error

Framework effectively simulates arbitrage-free volatility surface dynamics

Abstract

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.