= Solution
{c}
Let
$$
c(u)=\Phi\left(\frac{r+\sigma^2/2}{\sigma}\sqrt u\right)
-e^{-ru}\Phi\left(\frac{r-\sigma^2/2}{\sigma}\sqrt u\right),
\qquad c(0)=0.
$$
The <homogeneous function> of the <Black-Scholes formula> gives its at-the-money value at the start date as $S_{T_1}c(T-T_1)$. Before that date, <risk-neutral valuation> and the discounted stock <martingale> give
$$
e^{-r(T_1-t)}\mathbb E[S_{T_1}c(T-T_1)\mid\mathcal F_t]
=S_t c(T-T_1).
$$
After the start date the <strike price> is the realized value $S_{T_1}$, so the complete answer is
$$
\boxed{V_t=
\begin{cases}
S_t c(T-T_1),&0\leq t<T_1,\\
C(t,S_t;T,S_{T_1}),&T_1\leq t\leq T.
\end{cases}}
$$
At maturity the second line is $(S_T-S_{T_1})_+$. After entry the price depends on the historical <strike price> as well as the current stock, so that history cannot be discarded when giving the answer for all times.
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