First, what is a Brownian motion?

(i) 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 $W$ 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 a formula for the time-0 price of a European call option on the asset $S$ with strike $K$ and expiry $T$. You may use any general results, but should state them clearly.

(ii) In the same financial market, consider now a derivative which pays

$$Y=\left\{\exp \left(T^{-1} \int_{0}^{T} \log \left(S_{u}\right) d u\right)-K\right\}^{+}$$

at time $T$. Find the time-0 price for this derivative. Show that it is less than the price of the European call option which you derived in (i).