a) Compute the hedging strategy of the up-and-out barrier call option on the underlying asset price $S_{t}$ with exercise date $T$, strike price $K$ and barrier level $B$, with $B \geqslant K$.

b) Compute the joint probability density function

$$
\varphi_{Y_{T}, W_{T}}(a, b)=\frac{\mathrm{d} \mathbb{P}\left(Y_{T} \leqslant a \text { and } W_{T} \leqslant bight)}{d a d b}, \quad a, b \in \mathbb{R}
$$

of standard Brownian motion $W_{T}$ and its minimum

$$
Y_{T}=\min _{t \in[0, T]} W_{t}
$$

c) Compute the joint probability density function

$$
\varphi_{\breve{Y}_{T}, \widetilde{W}_{T}}(a, b)=\frac{\mathrm{d} \mathbb{P}\left(\breve{Y}_{T} \leqslant a \text { and } \widetilde{W}_{T} \leqslant bight)}{d a d b}, \quad a, b \in \mathbb{R}
$$

of drifted Brownian motion $\widetilde{W}_{T}=W_{T}+\mu T$ and its minimum

$$
\breve{Y}_{T}=\min _{t \in[0, T]} \widetilde{W}_{t}=\min _{t \in[0, T]}\left(W_{t}+\mu tight)
$$

d) Compute the price at time $t \in[0, T]$ of the down-and-out barrier call option on the underlying asset price $S_{t}$ with exercise date $T$, strike price $K$, barrier level $B$, and payoff

$$
C=\left(S_{T}-Kight)^{+} \mathbb{1}\left\{\min _{0 \leqslant t \leqslant T} S_{t}>Bight\}= \begin{cases}S_{T}-K & \text { if } \min _{0 \leqslant t \leqslant T} S_{t}>B \\ 0 & \text { if } \min _{0 \leqslant t \leqslant T} S_{t} \leqslant B\end{cases}
$$

in cases $0