Consider the price process \(\left(S_{t}ight)_{t \in[0, T]}\) given by

\[
\frac{d S_{t}}{S_{t}}=r d t+\sigma d B_{t}
\]

and a riskless asset valued \(A_{t}=A_{0} \mathrm{e}^{r t}, t \in[0, T]\), with \(r>0\). Compute the arbitrage-free price

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

at time \(t \in[0, T]\), of the \(\log\) contract with payoff \(\log S_{T}\).