  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.
