What is the effect of vol-of-vol of an underlying on the VIX Index?
The VIX is computed as hedging portfolio of log contracts to isolate pure volatility exposure without specifying an underlying stochastic process.
On the other hand, I can assume and parameterize a stochastic process, such as the Black-Scholes or Heston model, and compute a vol smile. Using that vol smile, I can compute the VIX, and should get the nonparametric implied vol.
In the BS-case, the result is trivial: The vol smile is flat, since the model assumes constant vol, and the VIX on that smile is exactly that. This is what I expect.
In the Heston-case, I am struggling: There is no pure vol parameter, just parameters for the vol process (initial vol, long-term vol, pullback, vol-of-vol, and correlation to the stock price process). Varying the parameters deforms the generated vol smile and affects the level of the VIX on that smile. The volatility of the Heston model can be computed using the moment formulas of the Fourier transform. I expected the VIX to be exactly that. This vol computation behaves as expected, increases with long-term vol, increases with vol-of-vol if rho is positive and vice versa, and so forth.
However: The VIX level completely ignores the vol-of-vol or correlation parameters.
If I understand correctly, the VIX is a zero delta portfolio of strike-weighted constant dollar gammas. Isn't the gamma itself sensitive to vol-of-vol changes? Why/Why not? Is there any way to hedge this?