This problem shows how the pricing formula for a T-year K-strike European cash-or-nothing call option of $1 can be retrieved from that of a plain vanilla T-year K-strike European call option.

(a) Using the fact that

for any real x, express the Black-Scholes price of the plain vanilla European call option in terms of the price(s) of appropriate cash-or-nothing call option(s). 

(b) Obtain the pricing formula for the cash-or-nothing call option by differentiating the Black-Scholes call price with respect to K.

(x-K)+ = K 1{a>c} de