In the context of the Black-Scholes model, let $S_{0}$ be the initial price of the stock, and let $\sigma$ be its volatility. Assume that the risk-free interest rate is zero and the stock pays no dividends. Let $\operatorname{EC}\left(S_{0}, K, \sigma, T\right)$ denote the initial price of a European call option with strike $K$ and maturity date $T$.

(a) Show that the Black-Scholes formula can be written in the form

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

where $d_{1}$ and $d_{2}$ depend on $S_{0}, K, \sigma$ and $T$, and $\Phi$ is the standard normal distribution function.

(b) Let $\operatorname{EP}\left(S_{0}, K, \sigma, T\right)$ be the initial price of a put option with strike $K$ and maturity $T$. Show that

$$\operatorname{EP}\left(S_{0}, K, \sigma, T\right)=\operatorname{EC}\left(S_{0}, K, \sigma, T\right)+K-S_{0}$$

(c) Show that

$$\operatorname{EP}\left(S_{0}, K, \sigma, T\right)=\operatorname{EC}\left(K, S_{0}, \sigma, T\right)$$

(d) Consider a European contingent claim with maturity $T$ and payout

$$S_{T} I_{\left\{S_{T} \leqslant K\right\}}-K I_{\left\{S_{T}>K\right\}}$$

Assuming $K>S_{0}$, show that its initial price can be written as $\mathrm{EC}\left(S_{0}, K, \hat{\sigma}, T\right)$ for a volatility parameter $\hat{\sigma}$ which you should express in terms of $S_{0}, K, \sigma$ and $T$.