Payout and profit analyses:

(a) A butterfly spread is the purchase of one call at exercise price X1, the sale of two calls at exercise price X2, and the purchase of one call at exercise price X3. X1 is less than X2, and X2 is less than X3 by equal amounts, and all calls have the same expiration date. Do the detailed payout and profit analysis of this strategy algebraically and graph them.

(b) A vertical combination is the purchase of a call with exercise price X2 and a put with exercise price X1, with X2 greater than X1. Do the detailed payout and profit analysis of this strategy algebraically and graph them.