Pricing Category ' $\mathrm{R}$ ' CBBC rebates. Given $\tau>0$, consider an asset price $\left(S_{t}ight)_{t \in[\tau, \infty)}$, given by

$$
S_{\tau+t}=S_{\tau} \mathrm{e}^{r t+\sigma W_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0
$$

where $\left(W_{t}ight)_{t \in \mathbb{R}_{+}}$is a standard Brownian motion, with $r \geqslant 0$ and $\sigma>0$. In what follows, $\Delta \tau$ is the deterministic length of the Mandatory Call Event (MCE) valuation period which commences from the time upon which a MCE occurs up to the end of the following trading session.

a) Compute the expected rebate (or residual) $\mathbb{E}\left[\left(\min _{s \in[0, \Delta \tau]} S_{\tau+s}-Kight)^{+} \mid \mathcal{F}_{\tau}ight]$ of a Category 'R' $C B B C$ Bull Contract having expired at a given time $\tauK>0$, with $r>0$.

b) Compute the expected rebate $\mathbb{E}\left[\left(\min _{s \in[0, \Delta \tau]} S_{\tau+s}-Kight)^{+} \mid \mathcal{F}_{\tau}ight]$ of a Category 'R' $C B B C$ Bull Contract having expired at a given time $\tau0$, with $r=0$.

c) Find the expression of the probability density function of the first hitting time

$$
\tau_{B}=\inf \left\{t \geqslant 0: S_{t}=Bight\}
$$

of the level $B>0$ by the process $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$.

d) Price the $\mathrm{CBBC}$ rebate

$$
\begin{aligned}
& \mathrm{e}^{-\Delta \tau} \mathbb{E}\left[\mathrm{e}^{-\tau} \mathbb{1}_{[0, T]}(\tau)\left(\min _{t \in[\tau, \tau+\Delta \tau]} S_{t}-Kight)^{+}ight] \\
& \quad=\mathrm{e}^{-\Delta \tau} \mathbb{E}\left[\mathrm{e}^{-r \tau} \mathbb{1}_{[0, T]}(\tau) \mathbb{E}\left[\left(\min _{t \in[\tau, \tau+\Delta \tau]} S_{t}-Kight)^{+} \mid \mathcal{F}_{\tau}ight]ight]
\end{aligned}
$$