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(a) Draw the payout profile for the following two call spread portfolios. (i) +1 K call and 1 (K + 1) call. (ii) +2 K

(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

Cx (t, T) |K+, ĈK where |K+ means the function is evaluated at K = K*.

(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.

k CK (t, T) |K*, where K* means the function is evaluated at K = K*.

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