For Questions assume all options are European style with maturity T.A'K call' is call option with strike price K. A 'knockout' option has payout zero if the defined event occurs.
Call spreads and digital options
(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
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.