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I was reading this great answer: What are the advantages/disadvantages of these approaches to deal with volatility surface?

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

This answer answers my question. But in the following equation from the answer

$C(t,S;T\to T+\theta,K) := E^Q[(\frac{S_{T+\theta}}{S_{T}}-K)_+] =: C_{BS}(S=1,\theta,K;\Sigma(t,S;T\to T+\theta,K))$

I don't understand why the payoff of the forward start option is not rather

$ E^Q[(S_{T+\theta}-S_{T})_+] $

and why it is set $S=1$

Joanna
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    Does this answer help? – AKdemy Sep 16 '21 at 20:34
  • @AKdemy thanks a lot! This helps a lot! But I still have an additional question, could you please have a look on the EDIT? – Joanna Sep 16 '21 at 21:15
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    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. – Quantuple Sep 17 '21 at 09:29

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