(i) What is Brownian motion?

(ii) Suppose that $\left(B_{t}\right)_{t \geqslant 0}$ is Brownian motion, and the price $S_{t}$ at time $t$ of a risky asset is given by

$$S_{t}=S_{0} \exp \left\{\sigma B_{t}+\left(\mu-\frac{1}{2} \sigma^{2}\right) t\right\}$$

where $\mu>0$ is the constant growth rate, and $\sigma>0$ is the constant volatility of the asset. Assuming that the riskless rate of interest is $r>0$, derive an expression for the price at time 0 of a European call option with strike $K$ and expiry $T$, explaining briefly the basis for your calculation.

(iii) With the same notation, derive the time-0 price of a European option with expiry $T$ which at expiry pays

$$\left\{\left(S_{T}-K\right)^{+}\right\}^{2} / S_{T}$$