(a) In the context of the Black-Scholes formula, let $S_{0}$ be the time- 0 spot price, $K$ be the strike price, $T$ be the time to maturity, and let $\sigma$ be the volatility. Assume that the interest rate $r$ is constant and assume absence of dividends. Write $\mathrm{EC}\left(S_{0}, K, \sigma, r, T\right)$ for the time- 0 price of a standard European call. The Black-Scholes formula can be written in the following form

$$\operatorname{EC}\left(S_{0}, K, \sigma, r, T\right)=S_{0} \Phi\left(d_{1}\right)-e^{-r T} K \Phi\left(d_{2}\right)$$

State how the quantities $d_{1}$ and $d_{2}$ depend on $S_{0}, K, \sigma, r$ and $T$.

Assume that you sell this option at time 0 . What is your replicating portfolio at time 0 ?

[No proofs are required.]

(b) Compute the limit of $\operatorname{EC}\left(S_{0}, K, \sigma, r, T\right)$ as $\sigma \rightarrow \infty$. Construct an explicit arbitrage under the assumption that European calls are traded above this limiting price.

(c) Compute the limit of $\operatorname{EC}\left(S_{0}, K, \sigma, r, T\right)$ as $\sigma \rightarrow 0$. Construct an explicit arbitrage under the assumption that European calls are traded below this limiting price.

(d) Express in terms of $S_{0}, d_{1}$ and $T$ the derivative

$$\frac{\partial}{\partial \sigma} \operatorname{EC}\left(S_{0}, K, \sigma, r, T\right)$$

[Hint: you may find it useful to express $\frac{\partial}{\partial \sigma} d_{1}$ in terms of $\frac{\partial}{\partial \sigma} d_{2}$.]

[You may use without proof the formula $S_{0} \Phi^{\prime}\left(d_{1}\right)-e^{-r T} K \Phi^{\prime}\left(d_{2}\right)=0$.]

(e) Say what is meant by implied volatility and explain why the previous results make it well-defined.