Assume that, under \(\mathbb{Q}, S\) follows a Black and Scholes dynamics with \(\sigma=1, r=0, S_{0}=1\). Prove that the function \(t \rightarrow C(1, t ; 1):=\) \(\mathbb{E}_{\mathbb{Q}}\left(\left(S_{t}-1\right)^{+}\right)\)is a cumulative distribution function of some r.v. \(X\); identify the law of \(X\).

\(\mathbb{E}_{\mathbb{Q}}\left(\left(S_{t}-1\right)^{+}\right)=\mathbb{Q}\left(4 B_{1}^{2} \leq t\right)\) where \(B\) is a \(\mathbb{Q}\)-BM.