Suppose that the dynamics of the risky asset are given by \(d S_{t}=S_{t}\left(b d t+\sigma(t) d B_{t}\right)\), where \(\sigma\) is a deterministic function. Characterize the law of \(S_{T}\) under the risk-neutral probability \(\mathbb{Q}\) and prove that the price of a European option on the underlying \(S\), with maturity \(T\) and strike \(K\), is \(\mathcal{B S}(x, \Sigma(t), t)\) where

 \((\Sigma(t))^{2}=\frac{1}{T-t} \int_{t}^{T} \sigma^{2}(s) d s\).