  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 and satisfactory fits of implied
volatility smiles for short maturities with very few parameters. Further, our
approach speeds up the calibration time and opens the door to time-efficient
simulation schemes.
