I'm reading the stochastic local volatility model literature, e.g., the Heston Stochastic Local Volatility model (https://ir.cwi.nl/pub/22747/22747D.pdf); but I'm a bit unsure about its calibration and pricing.
It seems to me that the calibration and pricing of an LSV model consist of the following steps:
- Use all the vanilla options to back out the local volatility surface (implied vol -> vol parametrization -> interpolation/extrapolation -> Dupire local vol).
- Choose whatever values you feel comfortable for the additional and exogenous model paramters, e.g., vov, mean-reverting speed etc.
- Generate MC paths to price whatever payoffs.
My question is about step #2: Since the Heston SLV is not a complete model, due to the fact that the additional and exogenous model parameters are not directly tradable, say, I know that for SPX vov, I can go to the VIX options market, but usually we don't have one for a single name. So, how do we decide the values for the model paramters? Do I go back and grab historical data to have a good guess and betting against the market?