American forward contracts. Consider (left(S_{t} ight)_{t in mathbb{R}_{+}})an asset price process given by [frac{d S_{t}}{S_{t}}=r d t+sigma

American forward contracts. Consider \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)an asset price process given by

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

where \(r>0\) and \(\left(B_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion under \(\mathbb{P}^{*}\).

a) Compute the price

\[f\left(t, S_{t}\right)=\operatorname{Sup}_{\substack{t \leqslant \tau \leqslant T \\ \tau \text { Stopping time }}} \mathbb{E}^{*}\left[\mathrm{e}^{-(\tau-t) r}\left(K-S_{\tau}\right) \mid S_{t}\right]\]

and optimal exercise strategy of a finite expiration American-type short forward contract with strike price \(K\) on the underlying asset priced \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\), with payoff \(K-S_{\tau}\) when exercised at time \(\tau \in[0, T]\).

b) Compute the price

\[f\left(t, S_{t}\right)=\operatorname{Sup}_{\substack{t \leqslant \tau \leqslant T \\ \tau \text { Stopping time }}} \mathbb{E}^{*}\left[\mathrm{e}^{-(\tau-t) r}\left(S_{\tau}-K\right) \mid S_{t}\right]\]

and optimal exercise strategy of a finite expiration American-type long forward contract with strike price \(K\) on the underlying asset priced \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\), with payoff \(S_{\tau}-K\) when exercised at time \(\tau \in[0, T]\).

c) How are the answers to Questions (a) and (b) modified in the case of perpetual options with \(T=+\infty\) ?