Briefly describe the Black-Scholes model. Consider a "cash-or-nothing" option with strike price $K$, i.e. an option whose payoff at maturity is

$$f\left(S_{T}\right)= \begin{cases}1 & \text { if } \quad S_{T}>K \\ 0 & \text { if } \quad S_{T} \leqslant K\end{cases}$$

It can be interpreted as a bet that the stock will be worth at least $K$ at time $T$. Find a formula for its value at time $t$, in terms of the spot price $S_{t}$. Find a formula for its Delta (i.e. its hedge ratio). How does the Delta behave as $t \rightarrow T$ ? Why is it difficult, in practice, to hedge such an instrument?