A butterfly spread consists of a long call with strike K − , two short calls with strikes K, and a long call with strike K + , for  > 0.

(a) Draw the payoff diagram of a butterfly spread. What is it a bet on?

(b) As explained in Section 16.1, selling straddles is a way to sell volatility, but it is clearly a high-risk strategy. To obtain insurance against the tail risks, we can add a long put and a long call. Draw the payoff diagram of a short straddle with strike K combined with a long put with strike K − 

and a long call with strike K + .

(c) What principle underlies the relation between the payoff diagrams in Parts

(a) and (b)? Explain.