Lifting the Heston model

How to reconcile the classical Heston model with its rough counterpart? We introduce a lifted version of the Heston model with n multi-factors, sharing the same Brownian motion but mean reverting at different speeds. Our model nests as extreme cases the classical Heston model (when n=1), and the rough Heston model (when n goes to infinity). We show that the lifted model enjoys the best of both worlds: Markovianity, satisfactory fits of implied volatility smiles for short maturities with very few parameters, and consistency with the statistical roughness of the realized volatility time series. Furthermore, our approach speeds up the calibration time and opens the door to time-efficient simulation schemes.