# Forward volatility smile: Local Volatility vs Stochastic volatility

And I have the following question:

How to show that the forward volatility smile of Local Volatility flattens while the one from Stochastic Volatility does not?

Could you please provide a mathematical demonstration of this fact?

EDIT

$$C(t,S;T\to T+\theta,K) := E^Q[(\frac{S_{T+\theta}}{S_{T}}-K)_+] =: C_{BS}(S=1,\theta,K;\Sigma(t,S;T\to T+\theta,K))$$
$$E^Q[(S_{T+\theta}-S_{T})_+]$$
and why it is set $$S=1$$
• It's just a question of absolute forward start and relative forward start. Relative forward start are more appealing in general since they have nice homogeneity properties under log-space invariant diffusion models (like BS or most SV models). In other words, the last equality of the first equation you mention (especially the $S=1$ part and keeping the relative moneyness $K$) can only be written under a homogeneous model such as BS. Sep 17, 2021 at 9:29