Consider an underlying asset price process written as

\[S_{t}=S_{0} \mathrm{e}^{r t+\sigma \widehat{B}_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]

where \(\left(\widehat{B}_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion under the risk-neutral probability measure \(\mathbb{P}^{*}\), with \(\sigma, r>0\).

a) Show that the processes \(\left(Y_{t}\right)_{t \in \mathbb{R}_{+}}\)and \(\left(Z_{t}\right)_{t \in \mathbb{R}_{+}}\)defined as

\[Y_{t}:=\mathrm{e}^{-r t} S_{t}^{-2 r / \sigma^{2}} \quad \text { and } \quad Z_{t}:=\mathrm{e}^{-r t} S_{t}, \quad t \geqslant 0\]

are both martingales under \(\mathbb{P}^{*}\).

b) Let \(\tau_{L}\) denote the hitting time \[\tau_{L}=\inf \left\{u \in \mathbb{R}_{+}: S_{u}=L\right\}\]
By application of the Stopping Time Theorem 14.7 to the martingales \(\left(Y_{t}\right)_{t \in \mathbb{R}_{+}}\)and \(\left(Z_{t}\right)_{t \in \mathbb{R}_{+}}\), show that \[\mathbb{E}^{*}\left[\mathrm{e}^{-r \tau_{L}} \mid S_{0}=x\right]= \begin{cases}\frac{x}{L}, & 0

c) Compute the price \(\mathbb{E}^{*}\left[\mathrm{e}^{-r \tau_{L}}\left(K-S_{\tau_{L}}\right)\right]\) of a short forward contract under the exercise strategy \(\tau_{L}\).

d) Show that for every value of \(S_{0}=x\) there is an optimal value \(L_{x}^{*}\) of \(L\) that maximizes \(L \mapsto \mathbb{E}\left[\mathrm{e}^{-r \tau_{L}}\left(K-S_{\tau_{L}}\right)\right]\).

e) Would you use the stopping strategy \[\tau_{L_{x}^{*}}=\inf \left\{u \in \mathbb{R}_{+}: S_{u}=L_{x}^{*}\right\}\]
as an optimal exercise strategy for the short forward contract with payoff \(K-S_{\tau}\) ?