Compute the arbitrage-free price

\[
C\left(t, S_{t}ight)=\mathrm{e}^{-(T-t) r} \mathbb{E}_{\alpha}\left[\left(S_{T}ight)^{2} \mid \mathcal{F}_{t}ight]
\]

at time \(t \in[0, T]\) of the power option with payoff \(\left(S_{T}ight)^{2}\) in the framework of the Bachelier (1900) model of Exercise 7.16.