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We propose a new model for the coherent forecasting of both the impliedvolatility surfaces and the underlying asset returns.In the spirit of Guyon andLekeufack (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\&P500), we first study how implied volatility can be predicted using the pasttrajectory of the underlying asset price. Our empirical study reveals that alarge part of the movements of the at-the-money-forward implied volatility forup to two years maturities can be explained using the past returns and theirsquares. Moreover, we show that up to four years of the past evolution of theunderlying price should be used for the prediction and that this feedbackeffect gets weaker when the maturity increases. Building on this new stylizedfact, we fit to historical data a parsimonious version of the SSVIparameterization (Gatheral and Jacquier, 2014) of the implied volatilitysurface relying on only four parameters and show that the two parameters rulingthe at-the-money-forward implied volatility as a function of the maturityexhibit a path-dependent behavior with respect to the underlying asset price.Finally, we propose a model for the joint dynamics of the implied volatilitysurface and the underlying asset price. The latter is modelled using a variantof the path-dependent volatility model of Guyon and Lekeufack and the former isobtained by adding a feedback effect of the underlying asset price onto the twoparameters ruling the at-the-money-forward implied volatility in theparsimonious SSVI parameterization and by specifying a hidden semi-Markovdiffusion model for the residuals of these two parameters and the two otherparameters. Thanks to this model, we are able to simulate highly realisticpaths of implied volatility surfaces that are arbitrage-free.

Implied volatility (also) is path-dependent