A digital call option with strike K and expiry time T is a derivative that will pay the holder a fixed amount of money M at time T if the price S_T of the underlying share at that time is greater than, or equal to, K. Otherwise, the option pays out nothing.

Similarly, a digital put option pays an amount M only if S_T is less than K at expiry.

(b) Sketch the payoffs of a digital call option and a digital put option as functions of S_T .

(c) Let C_t and P_t be the prices at some time t (earlier than T) of a digital call option and a digital put option respectively, each with the same underlying, K, T and M.

Prove the put-call parity relationship for digital options: C_t + P_t = M e^-r(T-t)

(You should assume that the underlying share pays no dividends, and that all market participants have access to a bank account with continuously-compounded interest rate r for both borrowing and lending. Be sure to justify every step of your proof carefully.)