In the Black-Scholes model the price $\pi(C)$ at time 0 for a European option of the form $C=f\left(S_{T}\right)$ with maturity $T>0$ is given by

$$\pi(C)=e^{-r T} \int_{-\infty}^{\infty} f\left(S_{0} \exp \left(\sigma \sqrt{T} y+\left(r-\frac{1}{2} \sigma^{2}\right) T\right)\right) \frac{1}{\sqrt{2 \pi}} e^{-y^{2} / 2} d y$$

(a) Find the price at time 0 of a European call option with maturity $T>0$ and strike price $K>0$ in terms of the standard normal distribution function. Derive the put-call parity to find the price of the corresponding European put option.

(b) The digital call option with maturity $T>0$ and strike price $K>0$ has payoff given by

$$C_{\mathrm{digCall}}= \begin{cases}1 & \text { if } S_{T} \geqslant K \\ 0 & \text { otherwise }\end{cases}$$

What is the value of the option at any time $t \in[0, T]$ ? Determine the number of units of the risky asset that are held in the hedging strategy at time $t$.

(c) The digital put option with maturity $T>0$ and strike price $K>0$ has payoff

$$C_{\text {digPut }}= \begin{cases}1 & \text { if } S_{T}<K \\ 0 & \text { otherwise. }\end{cases}$$

Find the put-call parity for digital options and deduce the Black-Scholes price at time 0 for a digital put.