Question on Call spreads and digital options: [Source: Intro to Quantitative Finance by Stephen Blyth, Chapter 7 Exercise 4]

(a) Draw the payout profile for the following two call spread portfolios.

(i) +1 K call and 1 (K + 1) call.

(ii) +2 K calls and 2 (K + 0.5) calls.

(b) By constructing a series of portfolios of call spreads and taking limits, prove that the price at time t of a digital call, with strike K and payout 1, is given by

(c) Write down the equivalent formula for a digital put option in terms of put prices.

(d) By examining the payout profile, derive a put-call parity relationship for the digital call and digital put.

CK (t, T) OK where IK means the function is evaluated at K K*