Forward start options (Rubinstein (1991)). A forward start European call option is an option whose holder receives at time \(T_{1}\) (e.g. your birthday) the value of a standard European call option at the money and with maturity \(T_{2}>T_{1}\). Price this birthday present at any time \(t \in\left[0, T_{1}ight]\), i.e. compute the price

\[
\mathrm{e}^{-\left(T_{1}-tight) r} \mathbb{E}^{*}\left[\mathrm{e}^{-\left(T_{2}-T_{1}ight) r} \mathbb{E}^{*}\left[\left(S_{T_{2}}-S_{T_{1}}ight)^{+} \mid \mathcal{F}_{T_{1}}ight] \mid \mathcal{F}_{t}ight]
\]

at time \(t \in\left[0, T_{1}ight]\), of the forward start European call option using the Black-Scholes formula

\[
\begin{aligned}
\operatorname{Bl}(x, K, \sigma, r, T-t)= & x \Phi\left(\frac{\log (x / K)+\left(r+\sigma^{2} / 2ight)(T-t)}{|\sigma| \sqrt{T-t}}ight) \\
& -K \mathrm{e}^{-(T-t) r} \Phi\left(\frac{\log (x / K)+\left(r-\sigma^{2} / 2ight)(T-t)}{|\sigma| \sqrt{T-t}}ight)
\end{aligned}
\]

\(0 \leqslant t