Same questions as in Exercise 15.14 , this time for the option with payoff (kappa-left(S_{tau} ight)^{p}) exercised

Same questions as in Exercise 15.14 , this time for the option with payoff \(\kappa-\left(S_{\tau}\right)^{p}\) exercised at time \(\tau\), with \(p>0\).

Data from Exercise 15.14

Let \(p \geqslant 1\) and consider a power put option with payoff \[\left(\left(\kappa-S_{\tau}\right)^{+}\right)^{p}= \begin{cases}\left(\kappa-S_{\tau}\right)^{p} & \text { if } S_{\tau} \leqslant \kappa \\ 0 & \text { if } S_{\tau}>\kappa\end{cases}\]
exercised at time \(\tau\), on an underlying asset whose price \(S_{t}\) is written as \[S_{t}=S_{0} \mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0,\]
where \(\left(B_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion under the risk-neutral probability measure \(\mathbb{P}^{*}, r \geqslant 0\) is the risk-free interest rate, and \(\sigma>0\) is the volatility coefficient.
Given \(L \in(0, \kappa)\) a fixed price, consider the following choices for the exercise of a put option with strike price \(\kappa\) :
i) If \(S_{t} \leqslant L\), then exercise at time \(t\).
ii) Otherwise, wait until the first hitting time \(\tau_{L}:=\inf \left\{u \geqslant t: S_{u}=L\right\}\), and exercise the option at time \(\tau_{L}\).

a) Under the above strategy, what is the option payoff equal to if \(S_{t} \leqslant L\) ?

b) Show that in case \(S_{t}>L\), the price of the option is equal to \[f_{L}\left(S_{t}\right)=(\kappa-L)^{p} \mathbb{E}^{*}\left[\mathrm{e}^{-\left(\tau_{L}-t\right) r} \mid S_{t}\right]\]

c) Compute the price \(f_{L}\left(S_{t}\right)\) of the option at time \(t\).
Recall that by (15.4) we have \(\mathbb{E}^{*}\left[\mathrm{e}^{-\left(\tau_{L}-t\right) r} \mid S_{t}=x\right]=(x / L)^{-2 r / \sigma^{2}}\) for \(x \geqslant L\).

d) Compute the optimal value \(L^{*}\) that maximizes \(L \mapsto f_{L}(x)\) for all fixed \(x>0\).
Observe that, geometrically, the slope of \(x \mapsto f_{L}(x)\) at \(x=L^{*}\) is equal to \(-p(\kappa-\) \(\left.L^{*}\right)^{p-1}\).

e) How would you compute the American put option price \[f\left(t, S_{t}\right)=\operatorname{Sup}_{\substack{\tau \geqslant t \\ \tau \text { Stopping time }}} \mathbb{E}^{*}\left[\mathrm{e}^{-(\tau-t) r}\left(\left(\kappa-S_{\tau}\right)^{+}\right)^{p} \mid S_{t}\right] ?\]