Forward contracts revisited. Consider a risky asset whose price \(S_{t}\) is given by \(S_{t}=S_{0} \mathrm{e}^{\sigma B_{t}+r t-\sigma^{2} t / 2}, t \geqslant 0\), where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion. Consider a forward contract with maturity \(T\) and payoff \(S_{T}-\kappa\).

a) Compute the price \(C_{t}\) of this claim at any time \(t \in[0, T]\).

b) Compute a hedging strategy for the option with payoff \(S_{T}-\kappa\).