Calibration and pricing with the Stochastic Local Volatility model

Question

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:

1. Use all the vanilla options to back out the local volatility surface (implied vol -> vol parametrization -> interpolation/extrapolation -> Dupire local vol).
2. Choose whatever values you feel comfortable for the additional and exogenous model paramters, e.g., vov, mean-reverting speed etc.
3. 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?

Answer 1

You need to calibrate the local volatility function contingent with the stochastic volatility parameters you chose if you want to be able to price back the calibration vanillas. The standard recipe for fx is as follows. Roughly calibrate Heston to a small set of vanilla options. Adjust the vol of vol parameter resulting from that calibration up or down to capture information from some first generation liquid exotic you are interested in (like double no touch options). Use the local volatility function to now fix the calibration so that you can now capture accurately some larger set of vanilla options. Price what you are interested in. The magic ingredients are the conditional expectation view of local volatility and the fwd equation for the distribution function.