# 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. 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

Let's say K=1.

If c=0.5, you get a shape like this (as you alluded to):

And for c=-0.5, you get this shape:

So does look like butterfly.

• Ok, I see that it looks similar to a butterfly, but it's not equivalent to a butterfly with long call K-c, long call K+c, and two short calls K, is it?
– dlp
May 8 '20 at 0:18
• There is a "recipe" for turning calls into puts: the Put Call Parity. If you apply this procedure to the two calls in this portfolio we will see what the equivalent portfolio without calls would be, which may (or may not) give some insight into what is going on here. May 8 '20 at 11:56
• I think the butterfly is meant in the sense of risk/reward profile as opposed to the constituent of the portfolio. If you view both in profit terms, then the profile is similar. May 8 '20 at 13:58