# Forward starting options concepts

Consider $$t_0, with $$t_0=0$$ (today date) and the standard payoff of a vanilla forward starting call option,

$$F_{t,T} = (S_T - S_t\cdot K)^+$$, with strike $$K$$.

If the price of this option is quoted today at $$t_0$$, then we can infer some kind of Black-Scholes implied volatility $$\sigma_{imp}(t_0, K, t, T)$$ for which the corresponding BS-price agrees with the market price (at $$t_0$$).

Now, denote the BS-implied volatility at time $$t$$ of a call option with the above payoff by $$\hat{\sigma}(t,T,K,S_t)$$. Obviously, from the stand point of $$t_0$$ this is unknown as the market quotes for date $$t$$ do not yet exist.

My question is how does $$\sigma_{imp}(t_0, K, t, T)$$ relate to the unknown $$\hat{\sigma}_{imp}(t,T,K,S_t(\omega)$$? Is the first just a proxy of the second?

I'm aware the answer might be obvious but I'm trying to convince myself and better understand the concepts in bibliography. Any references/easy to read papers that clarifies all the above is appreciated.