(a) In the context of the Black-Scholes formula, let $S_{0}$ be spot price, $K$ be strike price, $T$ be time to maturity, and assume constant interest rate $r$, volatility $\sigma$ and absence of dividends. Write down explicitly the prices of a European call and put,

$$E C\left(S_{0}, K, \sigma, r, T\right) \text { and } E P\left(S_{0}, K, \sigma, r, T\right)$$

Use $\Phi$ for the normal distribution function. [No proof is required.]

(b) From here on assume $r=0$. Keeping $T, \sigma$ fixed, we shorten the notation to $E C\left(S_{0}, K\right)$ and similarly for $E P$. Show that put-call symmetry holds:

$$E C\left(S_{0}, K\right)=E P\left(K, S_{0}\right)$$

Check homogeneity: for every real $\alpha>0$

$$E C\left(\alpha S_{0}, \alpha K\right)=\alpha E C\left(S_{0}, K\right)$$

(c) Show that the price of a down-and-out European call with strike $K<S_{0}$ and barrier $B \leqslant K$ is given by

$$E C\left(S_{0}, K\right)-\frac{S_{0}}{B} E C\left(\frac{B^{2}}{S_{0}}, K\right)$$

(d)

(i) Specialize the last expression to $B=K$ and simplify.

(ii) Answer a popular interview question in investment banks: What is the fair value of a down-and-out call given that $S_{0}=100, B=K=80, \sigma=20 \%, r=0, T=1$ ? Identify the corresponding hedge. [It may be helpful to compute Delta first.]

(iii) Does this hedge work beyond the Black-Scholes model? When does it fail?