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I am struggling to understand the link between two definitions of forward implied volatility. The well kwown model free forward implied volatility from $T_1$ to $T_2$ is defined as:

$$ \sigma_F(T_1, T_2, K)=\sqrt{\frac{\sigma(T_2,K)^2T_2-\sigma(T_1,K)^2T_1}{T_2-T_1}}. $$

In this paper, the forward implied volatility is implicitely defined as (p15):

$$ C_\mathrm{Model}(T_1,T_2, K) = e^{-rT_1}C_{\mathrm{BS}}(K, T_2-T_1, \sigma_F(T_1, T_2, K)). $$

Are both of them equivalent formulas ?

Thank you very much!

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No, the second one depends on hitting an arbitrary price. If that price is intrinsic, the the implied vol is 0.

First one depends on market information of today, which may imply something different in the future, there is no guarantee that the market is interested in getting the forward vol right when trading vanillas. It is only usable if underlying dynamics are lognormal for sure.

If in the second one, the model is BS, then I think it is the same as first.

If you can tell us how you want to use this, we can help you with more.

Thanks!

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