Derive the Black-Scholes formula $C\left(S_{0}, K, r, T, \sigma\right)$ for the time- 0 price of a European call option with expiry $T$ and strike $K$ written on an asset with volatility $\sigma$ and time-0 price $S_{0}$, and where $r$ is the riskless rate of interest. Explain what is meant by the delta hedge for this option, and determine it explicitly.

In terms of the Black-Scholes call option price formula $C$, find the time- 0 price of a forward-starting option, which pays $\left(S_{T}-\lambda S_{t}\right)^{+}$at time $T$, where $0<t<T$ and $\lambda>0$ are given. Find the price of an option which pays $\max \left\{S_{T}, \lambda S_{t}\right\}$ at time $T$. How would this option be hedged?