(a) Financial engineering deals with the design of new assets. Draw the payoff (at t=1) of the following bull butterfly spread:

Purchase 1 call with exercise price a

Sell 2 calls with exercise price (a+b)/2

Purchase 1 call with exercise price b

as a function of the underlying stock price S at t=1 where a=120 and b=140.

(b) An individual agent thinks that there is a high probability that the Dow Jones will have a payoff (or points) between a=23,000 and b=24,000 at t=1.

Design a digital option (see Figure 1) as a sequence of calls on the Dow that converges to a pure bet on getting $1 on the interval [23,000, 24,000], i.e. if the Dow lies between S[23,000, 24,000] at t=1, then the portfolio of calls pays off exactly $1. The payoff is 0 otherwise.