# Butterfly spread calls and puts

I am trying to understand the butterfly spread. My book (ASM Study Manual for SOA Investment & Financial Markets (IFM) Exam) says one of the ways to write it is:

Long put, strike $$=K-c$$

Short put, strike $$=K$$

Short call, strike $$=K$$

Long call, strike $$=K+c$$

When I try to calculate it, it doesn't look like a butterfly spread to me. There are four places where the stock price, $$S$$ could be. One of them is:

$$K+c > K > K-c > S$$

In this case payoff is

Long put $$max[0, (K-c)-S]= (K-c)-S$$

Short put, $$min[0, S-K]= S-K$$

Short call, $$min[0, K-S]= 0$$

Long call, $$max[0, S-(K+c)]= 0$$

The sum is $$-c$$, which is a negative payoff. I thought regular butterflies don't have negative payoffs? Is this a mistake? If so, is there a way to make a butterfly with both calls and puts rather than just one or the other?

• What is the title of the book? This does not look right to me either. – noob2 May 7 '20 at 22:11
• I think it's a mistake, I just want to confirm. Also, I want to know if there is a way to construct a butterfly with calls and puts, rather than with just one or the other. – dlp May 7 '20 at 22:14