Exercise 10.3 Consider a risky asset whose price $S_{t}$ is given by

$$
\begin{equation*}
d S_{t}=\sigma S_{t} d W_{t}+\frac{\sigma^{2}}{2} S_{t} d t \tag{10.25}
\end{equation*}
$$

where $\left(W_{t}ight)_{t \in \mathbb{R}_{+}}$is a standard Brownian motion.

a) Solve the stochastic differential equation (10.25).

b) Compute the expected stock price value $\mathbb{E}^{*}\left[S_{T}ight]$ at time $T$.

c) What is the probability distribution of the maximum $\underset{t \in[0, T]}{\operatorname{Max}} W_{t}$ over the interval $[0, T]$ ?

d) Compute the expected value $\mathbb{E}^{*}\left[M_{0}^{T}ight]$ of the maximum

$$
M_{0}^{T}:=\operatorname{Max}_{t \in[0, T]} S_{t}=S_{0} \operatorname{Max}_{t \in[0, T]} \mathrm{e}^{\sigma W_{t}}=S_{0} \exp \left(\sigma \underset{t \in[0, T]}{\operatorname{Max}} W_{t}ight)
$$

of the stock price over the interval $[0, T]$.

e) What is the probability distribution of the minimum $\min _{t \in[0, T]} W_{t}$ over the interval $[0, T]$ ?

f) Compute the expected value $\mathbb{E}^{*}\left[m_{0}^{T}ight]$ of the minimum

$$
m_{0}^{T}:=\min _{t \in[0, T]} S_{t}=S_{0} \min _{t \in[0, T]} \mathrm{e}^{\sigma W_{t}}=S_{0} \exp \left(\sigma \min _{t \in[0, T]} W_{t}ight)
$$

of the stock price over the interval $[0, T]$.