Implied volatility (also) is path-dependent

We propose a new model for the coherent forecasting of both the implied volatility surfaces and the underlying asset returns. In the spirit of Guyon and Lekeufack (2023) who are interested in the dependence of volatility indices (e.g. the VIX) on the paths of the associated equity indices (e.g. the S&P 500), we first study how implied volatility can be predicted using the past trajectory of the underlying asset price. Our empirical study reveals that a large part of the movements of the at-the-money-forward implied volatility for up to two years maturities can be explained using the past returns and their squares. Moreover, we show that up to four years of the past evolution of the underlying price should be used for the prediction and that this feedback effect gets weaker when the maturity increases. Building on this new stylized fact, we fit to historical data a parsimonious version of the SSVI parameterization (Gatheral and Jacquier, 2014) of the implied volatility surface relying on only four parameters and show that the two parameters ruling the at-the-money-forward implied volatility as a function of the maturity exhibit a path-dependent behavior with respect to the underlying asset price. Finally, we propose a model for the joint dynamics of the implied volatility surface and the underlying asset price. The latter is modelled using a variant of the path-dependent volatility model of Guyon and Lekeufack and the former is obtained by adding a feedback effect of the underlying asset price onto the two parameters ruling the at-the-money-forward implied volatility in the parsimonious SSVI parameterization and by specifying a hidden semi-Markov diffusion model for the residuals of these two parameters and the two other parameters. Thanks to this model, we are able to simulate highly realistic paths of implied volatility surfaces that are arbitrage-free.