(i) Suppose that the price $S_{t}$ of an asset at time $t$ is given by

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

where $B$ is a Brownian motion, $S_{0}$ and $\sigma$ are positive constants, and $r$ is the riskless rate of interest, assumed constant. In this model, explain briefly why the time-0 price of a derivative which delivers a bounded random variable $Y$ at time $T$ should be given by $E\left(e^{-r T} Y\right)$. What feature of this model ensures that the price is unique?

Derive an expression $C\left(S_{0}, K, T, r, \sigma\right)$ for the time- 0 price of a European call option with strike $K$ and expiry $T$. Explain the italicized terms.

(ii) Suppose now that the price $X_{t}$ of an asset at time $t$ is given by

$$X_{t}=\sum_{j=1}^{n} w_{j} \exp \left\{\sigma_{j} B_{t}+\left(r-\frac{1}{2} \sigma_{j}^{2}\right) t\right\}$$

where the $w_{j}$ and $\sigma_{j}$ are positive constants, and the other notation is as in part (i) above. Show that the time-0 price of a European call option with strike $K$ and expiry $T$ written on this asset can be expressed as

$$\sum_{j=1}^{n} C\left(w_{j}, k_{j}, T, r, \sigma_{j}\right)$$

where the $k_{j}$ are constants. Explain how the $k_{j}$ are characterized.