TD - Cash-or-nothing puts and calls, payoff diagrams and the Black-Scholes equation-Solution

Question
A cash-or-nothing call is an option which pays at expiry $1 at expiry T if the
spot price is above the strike K and nothing if S ≤ K. A cash-or-nothing
put is an option which pays out nothing if S > K and $1 if S ≤ K. Let Cb
and Pb denote the values of cash-or-nothing calls and puts respectively.
Assuming both options have the same expiry date T, derive the put-call
parity relation
Cb + Pb = e
−r(T −t)
.
By considering relevant payoff diagrams, show that the payoff for Cb is equivalent to the delta of a vanilla European call option, with the same strike, at
expiry. By differentiating the Black-Scholes equation with respect to S show
that the value of a cash-or-nothing call option on an underlying which pays
no dividend yield is equal to the delta of a vanilla European call option written on an underlying which pays a continuous dividend yield q
∗ and different
interest rate r
∗ and determine q
∗ and r
∗
.
Hence show that Cb is given by
Cb(S,t) = N(d2).