(i) What does it mean to say that $\left(S_{t}^{0}, S_{t}\right)_{0 \leqslant t \leqslant T}$ is a Black-Scholes model with interest rate $r$, drift $\mu$ and volatility $\sigma$ ?

(ii) Write down the Black-Scholes pricing formula for the time- 0 value $V_{0}$ of a time- $T$ contingent claim $C$.

(iii) Show that if $C$ is a European call of strike $K$ and maturity $T$ then

$$V_{0} \geqslant S_{0}-e^{-r T} K$$

(iv) For the European call, derive the Black-Scholes pricing formula

$$V_{0}=S_{0} \Phi\left(d^{+}\right)-e^{-r T} K \Phi\left(d^{-}\right)$$

where $\Phi$ is the standard normal distribution function and $d^{+}$and $d^{-}$are to be determined.

(v) Fix $t \in(0, T)$ and consider a modified contract which gives the investor the right but not the obligation to buy one unit of the risky asset at price $K$, either at time $t$ or time $T$ but not both, where the choice of exercise time is to be made by the investor at time $t$. Determine whether the investor should exercise the contract at time $t$.