How a Four-Leg Call-and-Put Butterfly Depends on Strike Spacing

Summary

The document considers a four-option position consisting of a long put below the middle strike, two short options at the middle strike, and a long call above it. The questioner calculates one region of the payoff and finds a negative amount, which appears inconsistent with the familiar nonnegative payoff of a standard butterfly.

The response points to the role of the spacing parameter: it illustrates payoff shapes for both a positive and a negative value of that parameter and says the resulting position does have a butterfly shape. This is a brief visual clarification, not a full payoff derivation. It does not show the diagrams in the supplied text, state the net premium or profit at expiration, or explain under what strike-ordering and sign conventions the payoff is nonnegative. Readers should distinguish the payoff shape from total trade profit, which also depends on the initial cost.

Key ideas

• A four-leg butterfly can combine puts below the middle strike and calls above it, with short options at the middle strike.
• The spacing parameter’s sign affects how the listed strikes are ordered.
• A negative payoff in one region does not by itself settle whether the overall payoff profile has a butterfly shape.
• The source gives a visual claim but does not provide the diagrams or a complete payoff derivation.

Tags

# Butterfly spread calls and puts


# 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?

## Answer by Magic is in the chain (score 1, accepted)

https://quant.stackexchange.com/a/53956

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.