(i) Give the definition of Brownian motion.

(ii) The price $S_{t}$ of an asset evolving in continuous time is represented as

$$S_{t}=S_{0} \exp \left(\sigma W_{t}+\mu t\right)$$

where $\left(W_{t}\right)_{t \geqslant 0}$ is a standard Brownian motion and $\sigma$ and $\mu$ are constants. If riskless investment in a bank account returns a continuously compounded rate of interest $r$, derive the Black-Scholes formula for the time-0 price of a European call option on asset $S$ with strike price $K$ and expiry $T$. [Standard results from the course may be used without proof but must be stated clearly.]

(iii) In the same financial market, a certain contingent claim $C$ pays $\left(S_{T}\right)^{n}$ at time $T$, where $n \geqslant 1$. Find the closed-form expression for the time- 0 value of this contingent claim.

Show that for every $s>0$ and $n \geqslant 1$,

$$s^{n}=n(n-1) \int_{0}^{s} k^{n-2}(s-k) d k .$$

Using this identity, how would you replicate (at least approximately) the contingent claim $C$ with a portfolio consisting only of European calls?