The following two graphs describe the payoff functions \(\phi\) of bull spread and bear spread options with payoff \(\phi\left(S_{N}ight)\) on an underlying asset priced \(S_{N}\) at maturity time \(N\).

Data from Figure 7.3

a) Show that in each case (i) and (ii) the corresponding option can be realized by purchasing and/or short selling standard European call and put options with strike prices to be specified.
b) Price the bull spread option in cases (i) and (ii) using the Black-Scholes formula.
An option with payoff \(\phi\left(S_{T}ight)\) is priced \(\mathrm{e}^{-r T} \mathbb{E}^{*}\left[\phi\left(S_{T}ight)ight]\) at time 0 . The payoff of the European call (resp. put) option with strike price \(K\) is \(\left(S_{T}-Kight)^{+}\), resp. \(\left(K-S_{T}ight)^{+}\).

Transcribed Image Text:

LA 50 100 K1 X 100 80 60 40 20 0 0 150 K2 200 100 80 60 40 20 0 0 50 K1 100 X 150 K2 (i) Bull spread payoff. (ii) Bear spread payoff. Figure 7.3: Payoff functions of bull spread and bear spread options. 200