Exercise 7.9 Butterfly options. A long call butterfly option is designed to deliver a limited payoff when the future volatility of the underlying asset is expected to be low. The payoff function of a long call butterfly option is plotted in Figure 7.5, with \(K_{1}:=50\) and \(K_{2}:=150\).

Data from figure 7.5

a) Show that the long call butterfly option can be realized by purchasing and/or issuing standard European call or put options with strike prices to be specified.
b) Price the long call butterfly option using the Black-Scholes formula.
c) Does the hedging strategy of the long call butterfly option involve holding or shorting the underlying stock?
Hints: Recall that an option with payoff \(\phi\left(S_{N}ight)\) is priced in discrete time as \((1+\) \(r)^{-N} \mathbb{E}^{*}\left[\phi\left(S_{N}ight)ight]\) at time 0 . The payoff of the European call (resp. put) option with strike price \(K\) is \(\left(S_{N}-Kight)^{+}\), resp. \(\left(K-S_{N}ight)^{+}\).

Transcribed Image Text:

60 50 40 30 20 10 0 0 50 K1 150 K2 Figure 7.5: Long call butterfly payoff function. 100 SN 200