# 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.

$$S(t_0)\left(\textbf{N}(d_1)-e^{-r(T_1-T_0)}\theta\textbf{N}(d_2)\right)$$ where $$d_1 = \frac{\ln(1/\theta)+(r+\sigma^2/2)(T_1-T_0)}{\sigma\sqrt{T_1-T_0}}, \hspace{1em} d_2 = \frac{\ln(1/\theta)+(r-\sigma^2/2)(T_1-T_0)}{\sigma\sqrt{T_1-T_0}},$$ $$\textbf{N}$$ is the cumulative distribution function of the standard normal distribution, $$\theta S(T_0)$$ is the strike price set at $$t=T_0$$, and $$t=T_1$$ is the time when the option matures.
In case it's of interest, it turns out that even in the Heston stochastic vol model we can find a semi-analytical formula for these options, for example the one given here: $$C_\text{FWS}(t,\nu(t),S(t)) = S(t)\hat{P}_1(t,\nu(t)) - ke^{-r(T-t^\ast)}S(t)\hat{P}_2(t,\nu(t))$$ with $$\hat{P}_j(t,\nu(t)) := \int_0^\infty P_j(t^\ast,1,\nu(t^\ast),k) \; f(\nu(t^\ast)|\nu(t))\text{ d}\nu(t^\ast)$$ where $$P_j$$ for $$j\in\{1,2\}$$ are the Heston probabilities given earlier in the paper and $$f(\nu(t^\ast)|\nu(t))=\frac{B}{2}\cdot e^{-(B\nu(t^\ast)+\Lambda)/2}\left(\frac{B\nu(t^\ast)}{\Lambda}\right)^{(R/2-1)/2}I_{R/2-1}\left(\sqrt{\Lambda B\nu(t^\ast)}\right).$$