  In industrial applications it is quite common to use stochastic volatility
models driven by semi-martingale Markov volatility processes. However, in order
to fit exactly market volatilities, these models are usually extended by adding
a local volatility term. Here, we consider the case of singular Volterra
processes, and we extend them by adding a local-volatility term to their Markov
lift by preserving the stylized results implied by these models on
plain-vanilla options. In particular, we focus on the rough-Heston model, and
we analyze the small time asymptotics of its implied local-volatility function
in order to provide a proper extrapolation scheme to be used in calibration.
